SOME PROBLEMS ON MANIFOLDS WITH LOWER BOUNDS ON
RICCI CURVATURE

By

Zhixin Wang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Mathematics-Doctor of Philosophy

2024

ABSTRACT

In this work, we delve into geometric analysis, particularly examining the interplay between lower

bounds on Ricci curvature and specific functionals. Our exploration begins with an investigation

into the implications of Yamabe invariants for asymptotically Poincaré-Einstein manifolds and their

conformal boundaries under conditions of 𝑅𝑖𝑐 ≥ −(𝑛 − 1)𝑔. We establish a relationship wherein

the type II Yamabe invariant of the conformal compactification of the manifold is bounded below

by the Yamabe invariant of its conformal boundary. Additionally, we focus on compact manifolds

with boundary where 𝑅𝑖𝑐 ≥ 0 and 𝐼 𝐼 ≥ 1, obtaining partial results concerning Wang’s conjecture.

Copyright by
ZHIXIN WANG
2024

ACKNOWLEDGEMENTS

This thesis represents not only my work at the keyboard but also a milestone in my life, made

possible with the support of several individuals.

First and foremost, I would like to express my deepest gratitude to my advisor, Xiaodong Wang,

whose expertise, understanding, and patience, added considerably to my graduate experience. I

appreciate his vast knowledge and skill in differential geometry. Without his guidance, this thesis

would not have been possible.

I would also like to thank my committee members, Kitagawa Jun, Thomas Parker, and Willie

Wong, for their insightful comments and suggestions. Their feedback significantly improved the

quality of this work.

I am deeply grateful to my family for their love, patience, and unwavering support.

I would also like to extend my heartfelt thanks to roommate Yuhan Jiang and friends from the

group SMDJ8, whose camaraderie and support have been instrumental throughout this journey.

Their understanding, laughter, and encouragement kept me grounded and motivated during the

most challenging times. Your friendship means the world to me, and I am truly grateful to have

each of you in my life.

This journey would not have been possible without all of you. Thank you.

iv

TABLE OF CONTENTS

CHAPTER 1

YAMABE INVARIANTS FOR ASMPTOTICALLY POINCARÉ-EINSTEIN
MANIFOLDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

1

CHAPTER 2

LIOUVILLE TYPE THEOREMS ON MANIFOLDS WITH LOWER
CURVATURE BOUND . . . . . . . . . . . . . . . . . . . . . . . . . . 27

CHAPTER 3

A LOG-SOBOLEV INEQUALITY . . . . . . . . . . . . . . . . . . . . 58

BIBLIOGRAPHY .

.

.

.

.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

v

CHAPTER 1

YAMABE INVARIANTS FOR ASMPTOTICALLY POINCARÉ-EINSTEIN MANIFOLDS

Roughly speaking, a Poincaré-Einstein manifold is a non-compact manifold characterized by nega-

tive constant Ricci curvature and the admission of a conformal compactification. The investigation

of Poincaré-Einstein manifolds is underpinned by a fundamental principle: the intricate interplay

between the manifold’s boundary and its interior. Given that we employ conformal transformations

in defining Poincaré-Einstein manifolds, a natural inquiry arises concerning the existence of con-

formal invariants that exemplify this principle. Such inequality was introduced in [CLW17] with

certain restrictions. Through collaborative efforts with X. Wang, we successfully eliminated these

constraints, resulting in a comprehensive and unrestricted conclusion [WW21], [WW22]. This

chapter will delve into the examination of these inequalities.

1.1 Asymptotically Poincaré-Einstein manifold

Poincaré-Einstein manifolds, which serve as the foundation for the AdS/CFT correspondence

framework , have been the subject of extensive research over the past three decades, yielding

significant advances in both mathematics and physics (see [Biq05], for instance).

The concept of the Poincaré-Einstein manifold emerges from an observation rooted in hyper-

bolic space (H𝑛, 𝑔𝐻). Utilizing the conformal ball model, this space can be effectively represented
(1−|𝑥|2)2 𝑑𝑥2), wherein 𝑑𝑥2 denotes the Euclidean metric. Through the application of the
, (H𝑛, 𝑔𝐻) can be conformally compactified to the unit disk within Eu-

as (B𝑛,
conformal factor (1−|𝑥|2)2

4

4

clidean space. The boundary of this compactified space is commonly termed the "boundary at

infinity" or the "conformal boundary." By summarizing this distinctive property in conjunction

with the Ricci curvature equation 𝑅𝑖𝑐𝑔𝐻 = −(𝑛 − 1)𝑔𝐻, we arrive at the comprehensive definition

of Poincaré-Einstein manifolds:

Definition 1.1.1. 𝑋 is the interior of a compact manifold ¯𝑋 with boundary 𝑀. (𝑋, 𝑔+) is called a
𝐶3,𝛼 Poincaré-Einstein manifold if 𝑔+ is a noncompact complete metric,

𝑅𝑖𝑐𝑔+ = −(𝑛 − 1)𝑔+

(1.1.1)

1

and 𝑔 = 𝜌2𝑔+ can be 𝐶3,𝛼 extended to ¯𝑋 by a boundary defining function 𝜌, i.e.

𝜌 ∈ 𝐶∞( ¯𝑋), 𝜌 > 0 in 𝑋, 𝜌 = 0 and d𝜌 ≠ 0 on 𝜕 𝑋.

𝜕 𝑋, together with the conformal class [𝜌2𝑔(cid:12)
If the Ricci curvature equation 1.1.1 is replaced by 𝑅𝑖𝑐𝑔+ = −(𝑛 − 1)𝑔+ + 𝑜(𝜌2), we arrive at the

(cid:12)𝜕 𝑋], is called conformal infinity.

definition for asymptotically Poincaré-Einstein manifolds.

Apart from hyperbolic space (B𝑛, 𝑔H), which serves as the prototype, Poincaré-Einstein mani-

folds also come in different ways.

Example 1.1.1. Perturbation from (B𝑛, 𝑔H) Let ℎ be the standard round metric on S𝑛−1. The work

by J.Lee and C.Graham showed that if we perturb the metric on S𝑛−1 slightly to ℎ′, then there exists

a corresponding 𝑔′ satisfying (1.1.1) and (S𝑛−1, ℎ′) as its conformal boundary. [GL91]

Example 1.1.2. Let (𝑁 𝑛−1, 𝑔𝑁 ) be a compact manifold without boundary, and 𝑅𝑖𝑐𝑁 = −(𝑛 − 2)𝑔𝑁 ,

then

is a Poincaré-Einstein manifold with compactification [0, 1] × 𝑁.

(R × 𝑁, 𝑑𝑡2 + cosh2(𝑡)𝑔𝑁 )

Note that the conformal boundary is 𝑁 × {±1}.

It has negative scalar curvature and is not

connected. We will revisit this example later, as it serves to illustrate how the conformal boundary

significantly influences the geometry of the entire manifold.

Given an asymptotically Poincaré-Einstein manifold, we want to study its geometry near con-

formal boundary. We start with the following result in [Lee94].

Theorem 1.1. Let (𝑋, 𝑔+) be asymptotically Poincaré-Einstein manifold with (𝑀, ℎ) as its confor-

mal boundary. For any ℎ′ ∈ [ℎ], there exists a boundary defining function 𝜌 so that near conformal

boundary 𝑔 takes the form

𝑔 =

1
𝜌2 (𝑑𝜌2 ⊕ ℎ𝜌)

(1.1.2)

where ℎ0 = ℎ′. In particular, |𝑑𝜌| 𝜌2𝑔+ = 1.

2

This is called Graham-Lee normal form. 𝜌 is a distance function for ¯𝑔, and its curvature can be

computed using Riccati equation, Gauss-Codazzi equation and Codazzi-Mainardi equations. Pick

local coordinates {𝑥𝑖} for 𝜕 𝑋 = 𝑀, and {𝑥0 = 𝜌, 𝑥𝑖} form local coordinates for 𝑋 near conformal

boundary. Apply (1.2.4), the traceless-Ricci curvature 𝐸 = 𝑅𝑖𝑐𝑔+ −
as

𝑅𝑔+
𝑛 𝑔+ are given in [BMW13]

2𝜌𝐸𝑖 𝑗 = −𝜌ℎ′′

𝑖 𝑗 + 𝜌ℎ 𝑝𝑞 ℎ′

𝑖 𝑝 ℎ′

𝑗 𝑞 −

𝜌
2

ℎ 𝑝𝑞 ℎ′

𝑝𝑞 ℎ′

𝑖 𝑗 + (𝑛 − 2)ℎ′

𝑖 𝑗 + ℎ 𝑝𝑞 ℎ′

𝑝𝑞 ℎ𝑖 𝑗 + 2𝜌𝑅𝑖𝑐(ℎ𝜌)𝑖 𝑗

𝐸𝑖0 =

1
2

𝐸00 = −

𝑝𝑞)

ℎ 𝑝𝑞 (∇𝑞 ℎ′
1
2

ℎ 𝑝𝑞 ℎ′′

𝑖 𝑝 − ∇𝑖 ℎ′
1
4

𝑝𝑞 +

ℎ 𝑝𝑞 ℎ𝑘𝑙 ℎ′

𝑝𝑘 ℎ′

𝑞𝑙 +

1
2𝜌

ℎ 𝑝𝑞 ℎ′

𝑝𝑞

(1.1.3)

where ′ denotes 𝜕

𝜕 𝜌 . Set 𝜌 = 0 in the first equation, and we get

(𝑛 − 2)ℎ′ + 𝑡𝑟ℎ (ℎ′)ℎ = 0

This implies ℎ′ = 0, and therefore 𝑀 is totally geodesic in ( ¯𝑋, ¯𝑔). In particular 𝑀 is umbilical
in ( ¯𝑋, ˜𝑔) for any conformal compactification ˜𝑔 since the property of umbilicus is invariant under

conformal change. Take derivative 𝜕

𝜕 𝜌 𝑘 times to the first equation of (1.1.3), and we get

(𝑛 − 1 − 𝑘)𝜕 𝑘

𝜌 ℎ + 𝑡𝑟ℎ (𝜕 𝑘

𝜌 ℎ)ℎ = 𝜕 𝑘−1

𝜌

(2𝜌𝐸)𝜌=0 + (terms containning 𝜕𝑙

𝜌 with

𝑙 < 𝑘)

Now suppose 𝐸 ≡ 0, i.e. (𝑀, 𝑔+) is Poincaré-Einstein. For 𝑘 < 𝑛 − 1, the coefficients for 𝜕 𝑘
𝜌 ℎ
is non-zero. By induction, we could solve for 𝜕 𝑘

𝜌 ℎ and thus get expansion for ¯𝑔 near conformal

boundary up to order 𝑛 − 2 if 𝑛 is even and 𝑛 − 1 if 𝑛 is odd. For example: i)𝜕 𝑘

𝜌 ℎ = 0 for 𝑘 odd and

𝑘 < 𝑛 − 1; ii) if 𝑛 is even, then 𝑡𝑟ℎ𝜕𝑛−1

𝜌 ℎ = 0 and 𝜕𝑛−1

𝜌 ℎ is not determined. (See Proposition 2.7 in

[Woo16], for example. The statement there is only for 𝑛 odd, but the argument works also for even

𝑛’s for orders below 𝑛 − 1). In particular, we can find the second order term

(cid:16)

− 2
𝑛−3

𝑅𝑖𝑐 (ℎ) − 𝑅ℎ

2(𝑛−2)

(cid:17)

ℎ

,

− 1

2 ℎ,

if 𝑛 ≥ 4;

if 𝑛 = 3.

ℎ′′ =





(1.1.4)

1.2 Conformal Invariants

In this section I will introduce basic formulas under conformal change and then introduce

Yamabe conformal invariants.

3

The Yamabe problem can be thought of as a continuation of uniformization theorem. For

2-dimensional spaces, all Riemannian surfaces are locally confomally Euclidean, and we have the

uniformization theorem

Theorem 1.2. Simply connected Riemann surface is biholomorphic to one of the following:

• ¯C = S2

• C

• {𝑧 ∈ C : |𝑧| < 1}

As a result, all compact Riemannian surfaces admit a conformal metric of constant Gaussian

curvature.

In dim>3, Weyl tensor is conformal invariant, thus obstruction for being locally conformally

flat. But we could still ask the following: can we find a metric of constant scalar curvature within

each conformal metric class. This is what Yamabe problem is about.

Given a Riemannian manifold (𝑀 𝑛, 𝑔) with 𝑛 ≥ 3 and local coordinates {𝑥𝑖}. Under conformal

change ¯𝑔 = 𝑢2𝑔, the new Levi-Civita connection can be calculated by

¯∇𝑋𝑌 = ∇𝑋𝑌 +

𝑋𝑢
𝑢

𝑌 +

𝑌 𝑢
𝑢

𝑋 −

𝑔(𝑋, 𝑌 )
𝑢

∇𝑢

(1.2.1)

The conformal change of Hession can be computed as

¯𝑔 (cid:0) ¯∇𝑋

¯∇ 𝑓 , 𝑌 (cid:1)

¯∇2 𝑓 (𝑋, 𝑌 ) = ¯𝑔 (cid:0) ¯∇𝑋
1
𝑢2 ∇ 𝑓 , 𝑌 (cid:1)
2𝑋𝑢
𝑢3 ∇ 𝑓 +
2𝑋𝑢
𝑢3 ∇ 𝑓 +
1
𝑢

= ∇2 𝑓 (𝑋, 𝑌 ) −

= ¯𝑔 (cid:0) −

= ¯𝑔

−

(cid:16)

¯∇𝑋 ∇ 𝑓 , 𝑌 (cid:1)

1
𝑢2
1
𝑢2 (∇𝑋 ∇ 𝑓 +
(𝑋𝑢 · 𝑌 𝑓 + 𝑋 𝑓 · 𝑌 𝑢) +

𝑋𝑢
𝑢

∇ 𝑓 +

(∇ 𝑓 )𝑢
𝑢

𝑋 −

𝑔(∇𝑢, ∇ 𝑓 )
𝑢

𝑋 𝑓
𝑢

(cid:17)

∇𝑢), 𝑌

𝑔(𝑋, 𝑌 )

4

which is

¯∇2 𝑓 (𝑋, 𝑌 ) = ∇2 𝑓 −

1
𝑢

(𝑑𝑢 ⊗ 𝑑𝑓 + 𝑑𝑓 ⊗ 𝑑𝑢) +

𝑔(∇𝑢, ∇ 𝑓 )
𝑢

𝑔

(1.2.2)

Using the formula above, the Riemannian curvature can be computed as

¯𝑅𝑖 𝑗 𝑘𝑙 = 𝑢2(𝑅𝑖 𝑗 𝑘𝑙 − 𝑔𝑖𝑘𝑇𝑗𝑙 + 𝑔 𝑗𝑙𝑇𝑖𝑘 − 𝑔𝑖𝑙𝑇𝑗 𝑘 − 𝑔 𝑗 𝑘𝑇𝑖𝑙)

where

𝑇𝑖 𝑗 =

∇𝑖∇ 𝑗 𝑢
𝑢

− 2

∇𝑖𝑢∇ 𝑗 𝑢
𝑢2

+

|𝑑𝑢|2
2𝑢2

𝑔𝑖 𝑗

Taking trace yields the formula for Ricci curvature and scalar curvature

¯𝑅𝑖 𝑗 = 𝑅𝑖 𝑗 − (𝑛 − 2)(

¯𝑅 =

(cid:16)

1
𝑢2

𝑅 − 2(𝑛 − 1)

∇𝑖∇ 𝑗 𝑢
𝑢
Δ𝑢
𝑢

− 2

∇𝑖𝑢∇ 𝑗 𝑢
𝑢2

) − (

− (𝑛 − 4) (𝑛 − 1)

+ (𝑛 − 3)

Δ𝑢
𝑢
|𝑑𝑢|2
𝑢2

(cid:17)

|𝑑𝑢|2
𝑢2 )𝑔𝑖 𝑗

For scalar curvature we usually take the form ¯𝑔 = 𝑢 4

𝑛−2 𝑔, and it takes the form

¯𝑅 = 𝑢− 𝑛+2

𝑛−2 (−

4(𝑛 − 1)
𝑛 − 2

Δ𝑢 + 𝑅𝑢)

(1.2.3)

(1.2.4)

Remark 1.2.1. For 𝑛 = 2, we use the conformal change ¯𝑔 = 𝑒2𝜙𝑔. The scalar curvature transforms

by

¯𝑅 = 𝑒−2𝜙 (−2Δ𝑢 + 𝑅)

(1.2.5)

The operator 𝐿𝑔 (𝑢) := −
conformal invariance. Let ¯𝑔 = 𝑢 4

4(𝑛−1)
𝑛−2 Δ𝑢 + 𝑅𝑢 is called conformal Laplacian.

It has the following

𝑛−2 𝑔. Suppose there is a third conformal metric 𝑔′ = 𝑣 4

𝑛−2 𝑔 =

( 𝑣
𝑢 )

4
𝑛−2 ¯𝑔. Then by (1.2.4)

𝑅′ = 𝑣− 𝑛+2

𝑛−2 𝐿𝑔 (𝑣) = (

𝑣
𝑢

⇒ 𝐿𝑔 (𝑣) = 𝑢 𝑛+2

𝑛−2 𝐿 ¯𝑔 (

𝑛−2 𝐿 ¯𝑔 (

)− 𝑛+2
𝑣
𝑢

)

𝑣
𝑢

)

(1.2.6)

Suppose 𝑋 𝑛 has a boundary Σ𝑛−1 and let 𝜈 be the outer normal vector. Under conformal change

¯𝑔 = 𝑢 4

𝑛−2 𝑔, the new normal vector becomes ¯𝜈 = 𝑢− 2

𝑛−2 𝜈. Using (1.2.1), the second fundamental

5

form 𝐼 𝐼 and mean curvature changes by

¯𝐼 𝐼 (𝑋, 𝑌 ) = 𝑢 2

𝑛−2 (cid:2)𝐼 𝐼 (𝑋, 𝑌 )) +

¯𝐻 = 𝑢− 2

𝑛−2 (𝐻 +

Now we can define Yamabe invariants.

𝜕𝑢
𝜕𝜈

𝑔(𝑋, 𝑌 )(cid:3)

(1.2.7)

2
(𝑛 − 2)𝑢
𝜕𝑢
𝜕𝜈

)

2(𝑛 − 1)
(𝑛 − 2)𝑢

Definition 1.2.1. Suppose (𝑀 𝑛, 𝑔) is a Riemannian manifold without boundary, the Yamabe in-

variant is defined to be

where

𝑌 (𝑀, [𝑔]) =

inf
𝑢∈𝐻1 (𝑀),𝑢≠0

𝐸𝑔 (𝑢)
𝑢 2𝑛

𝑛−2 dV)

𝑛−2
𝑛

(∫
𝑀

𝐸𝑔 (𝑢) =

∫

𝑀

4(𝑛 − 1)
𝑛 − 2

|∇𝑀𝑢|2 + 𝑅𝑀𝑢2dV

(1.2.8)

Remark 1.2.2. Pick a sequence of functions which blows up locally and it can be shown that

𝑌 (𝑀, [𝑔]) ≤ 𝑌 (S𝑛, 𝑑𝜃2)

(1.2.9)

where 𝑑𝜃2 is the round metric in S𝑛. See [Aub76].

If we write 𝑔′ = 𝑢 4

𝑛−2 𝑔 for 𝑢 > 0, the integral in (1.2.8) can be rewritten as

∫
𝑀

4(𝑛−1)
𝑛−2 |∇𝑔𝑢|2 + 𝑅𝑔𝑢2dV
(∫
𝑛−2 dV)
𝑀

𝑢 2𝑛

𝑛−2
𝑛

=

=

∫
𝑀

4(𝑛−1)
𝑛−2 Δ𝑔𝑢 + 𝑅𝑔𝑢)dV
𝑢(−
(∫
𝑢 2𝑛
𝑀
𝑅𝑔′dV𝑔′

𝑛−2 dV)

𝑛−2
𝑛

∫
𝑀

(Vol(𝑀, 𝑔′))

𝑛−2
𝑛

(1.2.10)

For general 𝑢 ∈ 𝐻1(𝑀), we can take |𝑢| and approximate it with positive functions in 𝐻1. An

equivalent definition for Yamabe invariant is thus derived

𝑌 (𝑀, [𝑔]) = inf
𝑔′∈[𝑔]

∫
𝑀

𝑅𝑔′dV𝑔′

(Vol(𝑀, 𝑔′))

𝑛−2
𝑛

where [𝑔] represents the conformal class of 𝑔. Derived from this definition, it becomes evident that

these two quantities remain invariant under conformal transformations, underscoring their pivotal

role in the realm of conformal geometry research.

6

The Euler-Lagrangian equation for (1.2.8) is given by

𝐿𝑔 (𝑢) = 𝜆𝑢 𝑛+2

𝑛−2

(1.2.11)

In conjunction with (1.2.4), the minimizer obtained from this equation provides a metric with

constant scalar curvature. Consequently, the existence of the minimizer resolves the problem

introduced at the beginning of this section. However, it is worth noting that in (1.2.8), we employ

the 𝐿 2𝑛

𝑛−2 (𝑋) norm in the denominator, and 2𝑛

𝑛−2 represents the critical power for Sobolev embedding.
While boundedness is assured, compactness is not guaranteed. To address this challenge, we employ

the “lowering index" technique.

We define a new functional as

𝑌𝑞 (𝑀, [𝑔]) =

inf
𝑢∈𝐻1 (𝑀),𝑢≠0

𝐸𝑔 (𝑢)

(cid:17) 2

𝑝

𝑢 𝑝d𝑉

(cid:16)∫
𝑀

(1.2.12)

where 𝑝 < 2𝑛

𝑛−2. These values of 𝑝 are strictly below the critical Sobolev conjugate. Using the
standard argument, the existence of minimizers 𝑢 𝑝 follows from the compactness of the inclusion

𝐻1(𝑀) ⊂ 𝐿 𝑝 (𝑀), and these 𝑢 𝑝’s satisfy the Euler-Lagrangian

𝐿𝑔 (𝑢) = 𝜆𝑢 𝑝−1

(1.2.13)

If we further impose the condition |𝑢| 𝑝 = 1, then 𝜆 𝑝 = 𝑌𝑞 (𝑀, [𝑔]) in (1.2.12). Similar for

(1.2.8). Trudinger[Tru68] and Aubin[Aub76] demonstrated that ∥𝑢 𝑝 ∥𝐿𝑟 is uniformly bounded

for some 𝑟 > 2𝑛

𝑛−2 provided the inequality in (1.2.9) is strict. Consequently, 𝑢 𝑝 converges to a
smooth solution 𝑢 of (1.2.11), and 𝑢 is a minimizer for (1.2.11). Thus, the primary challenge is

reduced to establishing the strict inequality in (1.2.9), except for standard spheres. This problem

was ultimately resolved by R. Schoen, who utilized the positive mass theorem to construct an

appropriate test function. Combining all the elements above, we arrive at the following theorem:

Theorem 1.3. Let (𝑀, 𝑔) be a compact manifold without boundary. Then𝑌 (𝑀, [𝑔]) ≤ 𝑌 (S𝑛, [𝑑𝜃2])

with inequality iff round metric on S. As a result, there exists a metric 𝑔′ ∈ [𝑔] such that 𝑅𝑔′ is

constant.

7

For a comprehensive exploration of this problem, refer to [LP87] or Chapter 5 of [SY94].

For manifolds with boundary (𝑀, Σ, 𝑔), we can ask the following two questions. Fix a conformal

class [𝑔], can we find 𝑔′ ∈ [𝑔] so that: I) 𝑅𝑔′ = constant, 𝐻𝑔′ = 0; or II) 𝑅𝑔′ = 0, 𝐻𝑔′ = constant.

These two are called Type I and Type II Yamabe problem respectively. As in Yamabe problem, we

can define the following two functional

Definition 1.2.2.

𝑌 (𝑋, 𝑀, [𝑔]) =

𝑄(𝑋, 𝑀, [𝑔]) =

inf
𝑢∈𝐻1,𝑢≠0

inf
𝑢∈𝐻1,𝑢≠0

(∫

𝑋

(∫
𝑀

𝐸 (𝑢)

𝑢2𝑛/(𝑛−2)) (𝑛−2)/𝑛

𝐸 (𝑢)

Type I

𝑢2(𝑛−1)/(𝑛−2)) (𝑛−2)/(𝑛−1)

Type II

(1.2.14)

where

𝐸 (𝑢) =

∫

𝑋

4(𝑛 − 1)
𝑛 − 2

|∇𝑢|2 + 𝑅𝑢2dV + 2

∫

𝑀

𝐻𝑢2dS

As before set 𝑔′ = 𝑢 4

𝑛−2 𝑔 for 𝑢 > 0, then
4(𝑛 − 1)
𝑛 − 2

|∇𝑢|2 + 𝑅𝑢2dV + 2

𝑢(−

4(𝑛 − 1)
𝑛 − 2

𝑅𝑔′dV𝑔′ + 2

Δ𝑢 + 𝑅𝑢)dV + 2

∫

Σ

𝐻𝑔′dS𝑔′

∫

𝑋
∫

𝑋
∫

𝑀

=

=

∫

𝐻𝑢2dS

𝑀
∫

𝑀

𝑢2(𝐻 +

2(𝑛 − 1)
𝑢(𝑛 − 2)

𝜕𝑢
𝜕𝜈

dS

And (1.2.14) can be rewritten as

𝑌 (𝑀, Σ, [𝑔]) = inf
𝑔′∈[𝑔]

𝑄(𝑀, Σ, [𝑔]) = inf
𝑔′∈[𝑔]

∫
𝑅𝑔′ + 2 ∫
𝐻𝑔′
𝑀
Σ
Vol(𝑀, 𝑔′) (𝑛−2)/𝑛

Type I

∫
𝑀

𝑅𝑔′ + 2 ∫
Σ
Area(Σ, 𝑔′|Σ) (𝑛−2)/(𝑛−1)

𝐻𝑔′

Type II

So these two minimum are conformal invariants. The corresponding Euler-Lagrangian equations

are computed to be

𝑇 𝑦 𝑝𝑒

𝐼 :

𝑇 𝑦 𝑝𝑒

𝐼 𝐼 :





𝐿𝑔 (𝑢) = 𝜆𝑢 𝑛+2

𝑛−2

𝜕𝑢




𝐿𝑔 (𝑢) = 0

𝜕𝜈 + 𝑛−2
2(𝑛−1)

𝐻𝑢 = 0

𝜕𝑢

𝜕𝜈 + 𝑛−2
2(𝑛−1)

𝐻𝑢 = 𝜆𝑢 𝑛

𝑛−2

8

(1.2.15)

(1.2.16)

So the minimizer of Type I and Type II Yamabe invariants solves the corresponding Yamabe

problems respectively by (1.2.4) and (1.2.7). Again, by picking suitable test functions we have

𝑄(𝑀, Σ, [𝑔]) ≤ 𝑄(B𝑛, S𝑛−1, [𝑑𝑥2])

𝑌 (𝑀, Σ, [𝑔]) ≤ ¯𝑄(S𝑛

+, S𝑛−1, [𝑑𝑠2])

(1.2.17)

And strict inequality implies the existence of minimizers by “lowering index" method. These

problems are only partially solved. For Type I Yamabe problem, the strict inequality was verified

in the following cases [Esc92b]

• 𝑛 = 3, 4, 5;

• 𝑛 ≥ 6 and 𝜕 𝑀 = Σ is not umbilic.

For Type II, Escobar verified the following in [Esc92a]

• 𝑛 > 6 and 𝑋 has a nonumbilic boundary point;

• 𝑛 ≥ 6, with 𝑋 locally flat and 𝜕 𝑋 unbilic;

• 𝑛 = 4, 5 and 𝜕 𝑋 is umbilic;

• 𝑛 = 3.

A substantial amount of work has been dedicated to addressing these two problems; nevertheless,

some cases still remain open. See, for instance, [Alm12], [Che09], [BC09], [Mar05], [Mar07], and

others. Recall that Poincaré-Einstein manifolds have umbilical boundaries. Apply these results and

direct arguments give us (see [CLW17])

Theorem 1.4. Let 𝑋 𝑛, 𝑔+ be 𝐶3,𝛼 Poincaré-Einstein manifold satisfying one of the following

• 3 ≤ 𝑛 ≤ 5

• 𝑛 ≥ 6 and 𝑋 is spin

9

Then there exists a conformal compactification ¯𝑔 = 𝜌2𝑔+ which is a minimizer for 𝑌 ( ¯𝑋, 𝑀, [ ¯𝑔]).

Furthermore, ¯𝑔 has constant scalar curvature and totally geodesic curvature.

Theorem 1.5. Let 𝑋 𝑛, 𝑔+ be 𝐶3,𝛼 Poincaré-Einstein manifold satisfying one of the following

• 3 ≤ 𝑛 ≤ 7

• 𝑛 ≥ 8 ad 𝑋 is spin

• 𝑛 ≥ 8 and 𝑋 is locally conformally flat

Then there exists a conformal compactification ¯𝑔 = 𝜌2𝑔+ which is a minimizer for 𝑄( ¯𝑋, 𝑀, [ ¯𝑔]).

Furthermore, ¯𝑔 has vanishing scalar curvature and constant mean curvature.

Remark 1.2.3. Some other approaches has been used to construct solutions to (1.2.16). Thus

solutions will provide us with metric of zero scalar curvature and constant mean curvature, but

they are not necessarily minimizers of Type II Yamabe invariant. See [Xu23].

1.3 A Sharp Inequality

Having established the Yamabe invariants in the previous section, we will now formulate

inequalities that establish a connection between the geometry of the boundary and the interior.

The work is initialized in [GH17]

Theorem 1.6. Let (𝑋, 𝑔+) be a Poincaré-Einstein manifold satisfying one of the following

𝑎)

3 ≤ 𝑛 ≤ 5,

or

𝑏) 𝑋 𝑖𝑠

𝑠𝑝𝑖𝑛.

Let ( ¯𝑋, 𝑀, ¯𝑔) be its compactification and ˆ𝑔 = ¯𝑔(cid:12)

(cid:12)𝑀. Then

𝑛
𝑛 − 2

𝑌 (𝑀, [ ˆ𝑔]) ≤ 𝑌 ( ¯𝑋, 𝑀, [ ¯𝑔])𝐼 ( ¯𝑋, 𝑋, ¯𝑔)2,

if

𝑛 ≥ 4

12𝜋 𝜒(𝑀) ≤ 𝑌 ( ¯𝑋, 𝑀, [ ¯𝑔])𝐼 ( ¯𝑋, 𝑋, ¯𝑔)2,

if

𝑛 = 3

where 𝐼 ( ¯𝑋, 𝑋, ¯𝑔) = Vol(𝑀, ˆ𝑔)1/(𝑛−1)/Vol( ¯𝑋, ¯𝑔)1/𝑛. Moreover, if the equality holds, then ¯𝑔 is

Einstein and ˆ𝑔 has constant scalar curvature.

10

This inequality tells in a certain sense that for Poincaré-Einstein manifolds, the conformal

geometry of the whole manifold can be controlled by the geometry of the conformal boundary.

While the inequality represents a significant breakthrough in Poincaré-Einstein manifold research,

it has limitations, notably that 𝐼 ( ¯𝑋, 𝑀, 𝑔) isn’t conformally invariant. In [CLW17], X. Chen, M.
Lai, and F. Wang introduced a new inequality (1.3.1) using 𝑄( ¯𝑋, 𝑀, 𝑔) instead of 𝑌 ( ¯𝑋, 𝑀, 𝑔).

Theorem 1.7. Let (𝑋, 𝑔+) be a Poincaré-Einstein manifold with compactification ( ¯𝑋, 𝑀, [ ¯𝑔]).

Suppose (𝑋, 𝑔+) satisfies one of the conditions in Thm1.5 then

𝑄( ¯𝑋, 𝑀, [ ¯𝑔]) ≥ 2

√︂ (𝑛 − 1)
𝑛 − 2

𝑌 (𝑀, 𝑔(cid:12)

(cid:12)𝑀) 𝑖 𝑓 𝑛 ≥ 4

𝑄( ¯𝑋, 𝑀, [ ¯𝑔]) ≥ 4√︁2𝜋 𝜒(𝑀) 𝑖 𝑓 𝑛 = 3

(1.3.1)

Moreover, the equality holds iff (𝑋, 𝑔+) is isometric to hyperbolic space (H𝑛, 𝑔H).

Sketch of proof By Thm1.5, the minimizer for 𝑄( ¯𝑋, 𝑀, [ ¯𝑔]) can be achieved. Say ¯𝑔 = 𝜌2𝑔+,

without loss of generality. Use 𝑅𝑖𝑐𝑔+ = −(𝑛 − 1)𝑔+ and (1.2.4) and traceless Ricci of ¯𝑔 is given by

¯𝐸 = −(𝑛 − 2) 𝜌−1 (cid:104) ¯∇2𝜌 −

(cid:105)

(Δ ¯𝑔 𝜌) ¯𝑔

1
𝑛

Integrating 𝜌| ¯𝐸 | ¯𝑔dV ¯𝑔 by parts yields

∫

¯𝑋

𝜌| ¯𝐸 | ¯𝑔dV ¯𝑔 =

∫

𝑀

1
𝜌

(cid:2)𝜕𝜈 | ¯∇𝜌|2

¯𝑔 +

1
𝜌

(1 − | ¯∇𝜌|2

¯𝑔)𝜕𝜈 𝜌(cid:3)dS ˆ𝑔

(1.3.2)

where 𝜈 is outer normal and ˆ𝑔 = ¯𝑔| 𝑀. Since ¯𝑔 has zero scalar curvature, by (1.2.4)

2𝜌Δ ¯𝑔 𝜌 = 𝑛(| ¯∇𝜌|2

¯𝑔 − 1)

By calculation in [Gra16], the equation above, together with (1.1.4) will give us local expansion

for 𝜌 near conformal boundary:

𝜕𝜈 𝜌 = 1,

𝜕2
𝜈 𝜌 = −

1
𝑛 − 1

¯𝐻,

𝜕3
𝜈 𝜌 =

1
𝑛 − 2

ˆ𝑅 −

1
𝑛 − 1

¯𝐻2

where ˆ𝑅 is the scalar curvature for ˆ𝑔 and ¯𝐻 is the mean curvature. Plug this into the integration

above, we get

2
(𝑛 − 2)2

∫

𝑋

𝜌| ¯𝐸 |2

¯𝑔dV ¯𝑔 =

∫

𝑀

11

(cid:16) 1
𝑛 − 2

¯𝐻2 −

1
𝑛 − 2

ˆ𝑅

(cid:17)dS ˆ𝑔

(1.3.1) follows by noting that ¯𝐻 is constant since ¯𝑔 minimizes Type II Yamabe invariant.

□

This inequality tells in a certain sense that for Poincaré-Einstein manifolds, the conformal

geometry of the whole manifold can be controlled by the geometry of the conformal boundary.

However, their findings were constrained by two primary limitations. Firstly, their work rested upon

the assumption that the minimizer of the second type Yamabe invariant could be realized. Further-

more, their approach was confined to Poincaré-Einstein manifolds, i.e. 𝑅𝑖𝑐𝑔+ = −(𝑛 − 1)𝑔+. It’s
important to note that many of the properties associated with Poincaré-Einstein manifolds extend

to asymptotically Poincaré-Einstein manifolds with 𝑅𝑖𝑐𝑔+ ≥ −(𝑛 − 1)𝑔+. The proof in [CLW17]
highly depends on the vanishing of traceless Ricci curvature, so their method fails in general setting.

In collaboration with X. Wang, we overcame these limitations, yielding the following result

[WW21][WW22]. This inequality highlights the intricate relationship between the manifold’s

boundary and its interior, aligning with our guiding principle.

Theorem 1.8. (𝑋, 𝑔+) asymptotically Poincaré-Einstein manifold with compactification ( ¯𝑋, 𝑀, ¯𝑔).

Suppose 𝑅𝑖𝑐𝑔+ ≥ −(𝑛 − 1)𝑔+ and the conformal infinity has nonnegative Yamabe invariant, then
√︂ (𝑛 − 1)
𝑛 − 2

𝑄( ¯𝑋, 𝑀, [ ¯𝑔]) ≥ 2

(cid:12)𝑀) 𝑖 𝑓 𝑛 ≥ 4

𝑌 (𝑀, ¯𝑔(cid:12)

(1.3.3)

𝑄( ¯𝑋, 𝑀, ¯𝑔) ≥ 4√︁2𝜋 𝜒(𝑀) 𝑖 𝑓 𝑛 = 3

Moreover, the equality holds iff (𝑋, 𝑔+) is isometric to hyperbolic space (H𝑛, 𝑔H).

Proof. The proof consists of three parts. First, we will define modified Yamabe quotients and

subsequently a quantity derived from it, playing a role analogous to 𝜌|𝐸 |2 in [CLW17]. Next,

we will analyze the asymptotic behavior of the function introduced in the initial step. Finally, we

will prove a sequence of inequalities for each modified Yamabe quotients, the limit of which will

yield the desired inequality. Finally we are going to prove rigidity, which in essence comes from

[CLW17].

Let ˆ𝑔 = ¯𝑔| 𝑀. Throughout the proof, operators and tensors with a + are defined with respect to 𝑔+,

those with a bar are defined with respect to ¯𝑔, and those with a hat are defined with respect to ˆ𝑔.

12

Step 1

From Cor1.1 in the next section, 𝑀 is connected. By Thm1.3 we can pick a ℎ ∈ [ ¯𝑔| 𝑀] so that 𝑅ℎ

is constant. Take Graham-Lee normal form (1.1.2). Lee [Lee94] constructed a function a positive

smooth function 𝜙 on 𝑋 s.t. Δ+𝜙 = 𝑛𝜙 and near 𝜕 𝑋

𝜙 = 𝜌−1 +

𝑅ℎ
4 (𝑛 − 1) (𝑛 − 2)

𝜌 + 𝑜

(cid:16)

𝜌2(cid:17)

He further proved that |𝑑𝜙|2

+ − 𝜙2 ≤ 0 in the following way. Since we assume 𝑌 (𝑀, [ℎ]) ≥ 0 and
𝑅ℎ is constant, 𝑅ℎ ≥ 0. By a direct calculation, |𝑑𝜙|2 − 𝜙2 has a continuation extension to 𝑀 and

|𝑑𝜙|2

+ − 𝜙2 ≤ 0 on 𝑀. By Bochner formula we have

Δ+(|𝑑𝜙|2

+ − 𝜙2) = 2𝑔+(∇+Δ+𝜙, ∇+𝜙) + 2|∇2

+𝜙|2

= 2𝑛|𝑑𝜙|2

+ + 2|∇2

+𝜙|2

+ + 2𝑅𝑖𝑐+(∇+𝜙, ∇+𝜙) −

= 2(𝑅𝑖𝑐+(∇+𝜙, ∇+𝜙) + (𝑛 − 1)|𝑑𝜙|2

+ + 2𝑅𝑖𝑐+(∇+𝜙, ∇+𝜙) − 2(𝜙Δ+𝜙 + |∇+𝜙|2
+)
+ − 2|∇+𝜙|2
|Δ+𝜙|2
1
𝑛

2
𝑛
+𝜙|2
+) + 2(|∇2

|Δ+𝜙|2)

+ −

+

≥ 0

As a result |𝑑𝜙|2

+ − 𝜙2 ≤ 0 on 𝑋. Consider the metric

𝑔 := 𝜙−2𝑔+ on ¯𝑋 . Its scalar curvature is
(cid:101)

given by

𝑅+ + 2 (𝑛 − 1) 𝜙−1Δ+𝜙 − 𝑛 (𝑛 − 1) 𝜙−2 |𝑑𝜙|2
+

(cid:101)𝑅 = 𝜙2 (cid:16)
≥ 𝜙2 (𝑅+ + 𝑛 (𝑛 − 1)) ≥ 0

(cid:17)

Moreover, by a direct calculation the boundary is totally geodesic. We consider the following

modified energy functional

˜𝐸 ( 𝑓 ) = 𝐸𝑔 ( 𝑓 ) −

∫

𝑋

(𝑅+ + 𝑛 (𝑛 − 1)) 𝜙2 𝑓 2𝑑𝑣𝑔.

(1.3.4)

Note that (𝑅 + 𝑛 (𝑛 − 1)) 𝜙2 ∈ 𝐶𝑚−3,𝛼 (cid:16)

𝑋

(cid:17) under our assumptions. More explicitly, by (1.3.4)

˜𝐸 ( 𝑓 ) =

∫

𝑋

(cid:20) 4 (𝑛 − 1)
𝑛 − 2

|𝑑𝑓 |2
𝑔 +
(cid:101)

(cid:16)

(cid:101)𝑅 − (𝑅 + 𝑛 (𝑛 − 1)) 𝜙2(cid:17)

(cid:21)

𝑓 2

𝑑𝑣𝑔 ≥ 0.

Since 𝑅+ + 𝑛 (𝑛 − 1) ≥ 0, we have

𝐸𝑔 ( 𝑓 ) ≥ (cid:101)𝐸 ( 𝑓 ) .

13

(1.3.5)

(1.3.6)

For 1 < 𝑞 ≤ 𝑛/(𝑛 − 2), consider

˜𝜆𝑞 := inf

(cid:101)𝐸 ( 𝑓 )
𝑀 | 𝑓 |𝑞+1 𝑑𝜎𝑔

(cid:16)∫

.

(cid:17) 2/(𝑞+1)

(1.3.7)

Lemma 1.3.1. Since ˜𝐸 ( 𝑓 ) ≥ 0, lim𝑞↗𝑛/(𝑛−2) (cid:101)𝜆𝑞 = (cid:101)𝜆𝑛/(𝑛−2).

Proof of lemma

Pick a minimizing sequence 𝑢𝑖 for 𝑄(𝑋, 𝑀, [ ¯𝑔]). For each 𝑢𝑖,
˜𝐸 (𝑢𝑖)

lim
𝑞↗ 𝑛
𝑛−2

(∫
𝑀

˜𝐸 (𝑢𝑖)
𝑢𝑞+1
𝑖

dS ˆ𝑔)

=

2
𝑞+1

(∫
𝑀

𝑢

2(𝑛−1)
𝑛−2

𝑖

dS ˆ𝑔)

𝑛−2
𝑛−1

As a result lim sup𝑞↗𝑛/(𝑛−2) (cid:101)𝜆𝑞 ≤ ˜𝜆𝑛/(𝑛−2). Since ˜𝐸 (𝑢) ≥ 0, by Hölder inequality

˜𝐸 (𝑢)
𝑢𝑞+1dS ˆ𝑔)

≥

2
𝑞+1

(∫
𝑀

(∫
𝑀

𝑢

˜𝐸 (𝑢)
2(𝑛−1)
𝑛−2 dS ˆ𝑔)

𝑛−2
𝑛−1

Area(𝑀, ˆ𝑔)

(𝑛−1) − 2
𝑛−2
𝑞+1

As a result ˜𝜆𝑞 ≥ ˜𝜆 𝑛

𝑛−2

Area(𝑀, ˆ𝑔)

(𝑛−1) − 2
𝑛−2

𝑞+1 . Take a limit, and we have

lim inf
𝑞↗𝑛/(𝑛−2)

(cid:101)𝜆𝑞 ≥ ˜𝜆𝑛/(𝑛−2)

□

Since ˜𝐸 ( 𝑓 ) ≥ 0, it is easy to see that lim𝑞↗𝑛/(𝑛−2) (cid:101)𝜆𝑞 = (cid:101)𝜆𝑛/(𝑛−2). Therefore, it suffices to prove

the above theorem for 𝑞 < 𝑛/(𝑛 − 2).
Since the trace operator 𝐻1 (cid:16)

𝑋

(cid:17)

→ 𝐿𝑞+1 (Σ) is compact for 𝑞 < 𝑛/(𝑛 − 2), by standard elliptic

theory, the above infimum 𝜆𝑞 is achieved by a smooth, positive function 𝑓 s.t.

and

∫

Σ

𝑓 𝑞+1𝑑𝜎 = 1

−

4(𝑛−1)
𝑛−2 Δ 𝑓 + 𝑅 𝑓 = (𝑅 + 𝑛 (𝑛 − 1)) 𝜙2 𝑓
𝜕 𝑓
𝜕𝜈 = 𝜆𝑞 𝑓 𝑞

4(𝑛−1)
𝑛−2

on 𝑋,

on 𝑀.

(1.3.8)

(1.3.9)





By the conformal invariance of the conformal Laplacian, we have

(cid:16)

𝑓 𝜙−(𝑛−2)/2(cid:17)

𝐿𝑔

= 𝜙−(𝑛+2)/2𝐿𝑔 ( 𝑓 )

= (𝑅 + 𝑛 (𝑛 − 1)) 𝑓 𝜙−(𝑛−2)/2.

14

In other words, 𝑢 := 𝑓 𝜙−(𝑛−2)/2 satisfies the following equation

−Δ𝑔+

𝑢 =

𝑛 (𝑛 − 2)
4

𝑢.

(1.3.10)

Write 𝑢 = 𝑣−(𝑛−2)/2. Then

Equivalently Δ𝑔+

𝑣 − 𝑛𝑣 = 𝑛

𝑣 =

𝑛
Δ𝑔+
2
2 Φ with Φ = 𝑣−1 (cid:16)

𝑣−1 (cid:16)

|𝑑𝑣|2
𝑔+

|𝑑𝑣|2
𝑔+
− 𝑣2(cid:17).

+ 𝑣2(cid:17)

.

Lemma 1.3.2. We have

(cid:16)

div+

𝑣−(𝑛−2)∇+Φ

(cid:17)

= 2𝑣−(𝑛−2)𝑄,

(1.3.11)

where

𝑄 =

(cid:12)
(cid:12)
(cid:12)
(cid:12)

∇2
+𝑣 −

Δ+𝑣
𝑛

𝑔+

(cid:12)
(cid:12)
(cid:12)
(cid:12)

2

+

+ 𝑅𝑖𝑐+ (∇+𝑣, ∇+𝑣) + (𝑛 − 1) |∇+𝑣|2

+ ≥ 0.

All the computation is done with respect to 𝑔+, but we drop the subscript to simplify the presentation.

Proof. As 𝑣Φ = |∇+𝑣|2 − 𝑣2, we have, by using the Bochner formula

1
2

(𝑣Δ+𝜙 + 2 ⟨∇+𝑣, ∇𝜙⟩+ + 𝜙Δ+𝑣) = (cid:12)

+ + ⟨∇+𝑣, ∇+Δ+𝑣⟩+ + 𝑅𝑖𝑐+ (∇+𝑣, ∇+𝑣) − 𝑣Δ+𝑣 − |∇+𝑣|2

+

2

+𝑣(cid:12)
(cid:12)∇2
(cid:12)
(Δ+𝑣)2
𝑛
Δ+𝑣
𝑛

=

=

=

+ ⟨∇+𝑣, ∇+Δ+𝑣⟩+ + 𝑣Δ+𝑣 − 𝑛 |∇+𝑣|2

+ + 𝑄

(Δ+𝑣 − 𝑛𝑣) + ⟨∇+𝑣, ∇+ (Δ+𝑣 − 𝑛𝑣)⟩+ + 𝑄

1
2

ΦΔ+𝑣 +

𝑛
2

⟨∇+𝑣, ∇+Φ⟩+ + 𝑄

Thus,

or

ΔΦ+ = (𝑛 − 2) 𝑣−1 ⟨∇+𝑣, ∇+Φ⟩+ + 2𝑄

(cid:16)

div+

𝑣−(𝑛−2)∇+Φ

(cid:17)

= 2𝑣−(𝑛−2)𝑄 ≥ 0.

div (cid:16)

𝑣−(𝑛−2)∇Φ

(cid:17) plays the role of traceless Ricci 𝐸 in [CLW17].

15

□

Step 2
In this part we are going to figure out the asymptotical expansion for terms in div (cid:16)

𝑣−(𝑛−2)∇Φ

(cid:17). We

now consider the metric 𝑔 = 𝑢4/(𝑛−2)𝑔+. Since 𝑢 = 𝑓 𝜙−(𝑛−2)/2, we also have

𝑔 = 𝑓 4/(𝑛−2) 𝜙−2𝑔+ = 𝑓 4/(𝑛−2)

𝑔.
(cid:101)

As 𝜕 𝑋 is totally geodesic w.r.t.

𝑔 and 𝑔 is conformal to
(cid:101)

𝑔, we know that 𝜕 𝑋 is umbilic w.r.t. 𝑔 and
(cid:101)

its mean curvature, in view of the boundary condition of (1.3.9), is given by

𝐻 =

𝜆𝑞
2

𝑓 𝑞− 𝑛

𝑛−2 .

(1.3.12)

Set 𝜌 = 𝑢2/(𝑛−2) = 𝑣−1. By a direct calculation, the equation (1.3.10) becomes, using 𝑔 as the

background metric

2𝜌Δ𝜌 = 𝑛

(cid:16)

|∇𝜌|2 − 1(cid:17)

.

(1.3.13)

Let 𝑡 be the geodesic distance to Σ w.r.t. 𝑔. We need the following lemma which is essentially

contained in [CLW17].

Lemma 1.3.3. Near Σ = 𝜕 𝑋, we can write

𝑔 = 𝑑𝑡2 + 𝑔𝑖 𝑗 (𝑡, 𝑥) 𝑑𝑥𝑖𝑑𝑥 𝑗 ,

where {𝑥1, · · · , 𝑥𝑛−1} are local coordinates on Σ. Then

𝜌 = 𝑡 −

𝐻
2 (𝑛 − 1)

𝑡2 +

1
6

(cid:18) 𝑅Σ
𝑛 − 2

−

(cid:19)

𝐻2
𝑛 − 1

𝑡3 + 𝑜

(cid:16)

𝑡3(cid:17)

.

In particular,

𝜕
𝜕𝜈

(cid:104)

𝜌−1 (cid:16)

|∇𝜌|2 − 1(cid:17)(cid:105)

|Σ =

𝑅Σ
𝑛 − 2

−

𝐻2
𝑛 − 1

.

Proof. For completeness, we present the proof showing that the Einstein condition is not required.

In local coordinates

|∇𝜌|2 =

Δ𝜌 =

(cid:19) 2

(cid:18) 𝜕 𝜌
𝜕𝑡
𝜕2𝜌
𝜕𝑡2 +

+ 𝑔𝑖 𝑗 𝜕 𝜌
𝜕𝑥𝑖
√
𝐺

𝜕 log
𝜕𝑡

𝜕 𝜌
𝜕𝑥 𝑗
𝜕 𝜌
𝜕𝑡

,

+

1
√

𝐺

𝜕
𝜕𝑥𝑖

(cid:18)

𝑔𝑖 𝑗 √
𝐺

(cid:19)

.

𝜕 𝜌
𝜕𝑥 𝑗

16

Restricting (1.3.13) on Σ on which both 𝜌 and 𝑟 vanish with order 1 yields 𝜕 𝜌

𝜕𝑡 |Σ = 1.

Differentiating (1.3.13) in 𝑡 yields

2
𝑛

(cid:18) 𝜕 𝜌
𝜕𝑡

Δ𝜌 + 𝜌

(cid:19)

Δ𝜌

𝜕
𝜕𝑡

= 2

𝜕 𝜌
𝜕𝑡

Evaluating both sides on Σ yields

𝜕2𝜌

𝜕𝑡2 + 2𝑔𝑖 𝑗 𝜕2𝜌

𝜕𝑥𝑖𝜕𝑡

𝜕 𝜌
𝜕𝑥 𝑗

− 𝑔𝑖𝑘 𝑔 𝑗𝑙 𝜕𝑔𝑘𝑙
𝜕𝑡

𝜕 𝜌
𝜕𝑥𝑖

𝜕 𝜌
𝜕𝑥 𝑗

.

(1.3.14)

(cid:32) 𝜕2𝜌
𝜕𝑡2 +

2
𝑛

√

𝐺

𝜕 log
𝜕𝑡

(cid:33)

|Σ = 2

𝜕2𝜌
𝜕𝑡2 |Σ.

Thus

𝜕2𝜌
𝜕𝑡2 |Σ =

1
𝑛 − 1

√

𝐺

𝜕 log
𝜕𝑡

|Σ = −

𝐻
𝑛 − 1

.

Differentiating the formula for Δ𝜌 we get
(cid:32) 𝜕3𝜌
𝜕𝑡3 +
(cid:32) 𝜕3𝜌
𝜕𝑡3 +

Δ𝜌|Σ =

𝜕
𝜕𝑡

=

√

𝐺

𝜕2 log
𝜕𝑡2

√

𝐺

𝜕2 log
𝜕𝑡2

+

+

√

𝐺

𝜕 log
𝜕𝑡

(cid:33)

𝜕2𝜌
𝜕𝑡2

|Σ

(cid:33)

𝐻2
𝑛 − 1

|Σ

Differentiating (1.3.14) in 𝑟 and evaluating on Σ, we obtain

2
𝑛

(cid:18) 𝜕2𝜌
𝜕𝑡2 Δ𝜌 + 2

𝜕
𝜕𝑡

(cid:19)

Δ𝜌

|Σ = 2

(cid:19) 2

(cid:18) 𝜕2𝜌
𝜕𝑡2

|Σ + 2

𝜕3𝜌
𝜕𝑡3 |Σ =

2𝐻2
(𝑛 − 1)2 + 2

𝜕3𝜌
𝜕𝑡3 |Σ.

Using the previous formulas, we arrive at

𝜕3𝜌
𝜕𝑡3 |Σ =

2
𝑛 − 2

(cid:32) 𝐻2
𝑛 − 1

+

By a direct calculation, we also have

√

𝐺

𝜕2 log
𝜕𝑡2

(cid:33)

|Σ

.

√

𝐺

𝜕2 log
𝜕𝑡2

|Σ = −𝑅𝑖𝑐 (𝜈, 𝜈) −

𝐻2
𝑛 − 1

.

Therefore

𝜕3𝜌
𝜕𝑡3 |Σ = −

2
𝑛 − 2
𝑅Σ
𝑛 − 2

−

=

𝑅𝑖𝑐 (𝜈, 𝜈)

𝐻2
𝑛 − 1

,

where we used the Gauss equation in the last step.

The second identity follows from a direct calculation.

□

17

Step 3

In this part we will use lemma1.3.3 in (1.3.11) to get the main result.

Integrating the identity

(1.3.11) on 𝑋𝜀 = {𝑡 ≥ 𝜀} yields

∫

2

𝑋𝜀

𝑣−(𝑛−2)𝑄𝑑𝑣𝑔+ =

∫

𝜕 𝑋𝜀

𝑣−(𝑛−2) 𝜕Φ
𝜕𝜈

𝑑𝜎𝑔+

.

Since 𝑔+ = 𝜌−2𝑔, we obtain by a direct calculation

∫

𝜕 𝑋𝜀

𝑣−(𝑛−2) 𝜕Φ
𝜕𝜈+

𝑑𝜎𝑔+ =

∫

𝜕 𝑋𝜀

𝜕
𝜕𝜈

(cid:104)

𝜌−1 (cid:16)

|∇𝜌|2 − 1(cid:17)(cid:105)

𝑑𝜎𝑔.

Therefore

∫

2

𝑋𝜀
Letting 𝜀 → 0, we obtain, in view of Lemma 1.3.3

𝜕 𝑋𝜀

𝑣−(𝑛−2)𝑄𝑑𝑣𝑔+ =

∫

𝜕
𝜕𝜈

(cid:104)

𝜌−1 (cid:16)

|∇𝜌|2 − 1(cid:17)(cid:105)

𝑑𝜎𝑔.

∫

2

𝑋

𝑣−(𝑛−2)𝑄𝑑𝑣𝑔+ =

∫

Σ

(cid:18) 𝑅Σ
𝑛 − 2

−

(cid:19)

𝐻2
𝑛 − 1

𝑑𝜎𝑔

(1.3.15)

The rest of the argument is the same as in [WW21]. We present it for completeness. By

(1.3.12) and the Holder inequality again

∫

Σ

𝐻2𝑑𝜎 =

=

≤

=

(cid:18) 𝜆𝑞
2
(cid:18) 𝜆𝑞
2

(cid:18) 𝜆𝑞
2
(cid:18) 𝜆𝑞
2

(cid:19) 2 ∫

Σ
(cid:19) 2 ∫

Σ
(cid:19) 2 (cid:18)∫

𝑓 2(𝑞− 𝑛

𝑛−2) 𝑓 2(𝑛−1)/(𝑛−2) 𝑑𝜎

𝑓 2(𝑞− 1

𝑛−2) 𝑑𝜎

(cid:19) 2(𝑞− 1

𝑛−2 )/(𝑞+1)

𝑓 𝑞+1𝑑𝜎

𝑉 (Σ, 𝑔) ( 𝑛

𝑛−2 −𝑞)/(𝑞+1)

Σ

(cid:19) 2

𝑉 (Σ, 𝑔)( 𝑛

𝑛−2 −𝑞)/(𝑞+1) .

Plugging the above inequality into (1.3.15), we obtain

∫

2

𝑋

𝑣−(𝑛−2)𝑄𝑑𝑣𝑔+ ≤

𝜆2
𝑞
4 (𝑛 − 1)

𝑉 (Σ, 𝑔) ( 𝑛

𝑛−2 −𝑞)/(𝑞+1) −

1
𝑛 − 2

∫

Σ

𝑅Σ𝑑𝜎.

(1.3.16)

When 𝑛 = 3, this implies

𝑞𝑉 (Σ, 𝑔) (3−𝑞)/(𝑞+1) ≥ 32𝜋 𝜒 (Σ) .
𝜆2

18

In the following, we assume 𝑛 > 3 . By (1.3.8) and the Hölder inequality

∫

1 =

𝑓 𝑞+1𝑑𝜎

Σ
(cid:18)∫

Σ

≤

𝑓 2(𝑛−1)/(𝑛−2) 𝑑𝜎

(cid:19) (𝑞+1) (𝑛−2)

2(𝑛−1)

𝑉 (Σ, 𝑔)

𝑛−𝑞 (𝑛−2)
2(𝑛−1)

= 𝑉 (Σ, 𝑔)

(𝑞+1) (𝑛−2)
2(𝑛−1) 𝑉 (Σ, 𝑔)

𝑛−𝑞 (𝑛−2)
2(𝑛−1)

Thus

𝑉 (Σ, 𝑔)−

𝑛−𝑞 (𝑛−2)
(𝑛−2) (𝑞+1) ≤ 𝑉 (Σ, 𝑔) .

Plugging this inequality into (1.3.16) yields

𝑣−(𝑛−2)𝑄𝑑𝑣𝑔+
(cid:34)

∫

2

𝑋
𝑛−1
𝑉 (Σ, 𝑔)
𝑛−3
4 (𝑛 − 1)
𝑛−1
𝑉 (Σ, 𝑔)
𝑛−3
4 (𝑛 − 1)

(cid:101)𝜆2
𝑞𝑉 (Σ, 𝑔)

2(𝑛−𝑞 (𝑛−2) )
(𝑛−3) (𝑞+1) −

(cid:20)

(cid:101)𝜆2
𝑞𝑉 (Σ, 𝑔)

2(𝑛−𝑞 (𝑛−2) )
(𝑛−3) (𝑞+1) −

4 (𝑛 − 1)

𝑛−1
𝑛−3

(𝑛 − 2) 𝑉 (Σ, 𝑔)
4 (𝑛 − 1)
(𝑛 − 2)

𝑌 (Σ, [𝛾])

(cid:35)

𝑅Σ𝑑𝜎

∫

Σ

.

(cid:21)

≤

≤

Therefore

(cid:101)𝜆2
𝑞 ≥

4 (𝑛 − 1)
(𝑛 − 2)

𝑌 (Σ) 𝑉 (Σ, 𝑔)−

2(𝑛−𝑞 (𝑛−2) )
(𝑛−3) (𝑞+1)

.

Finally let 𝑞 ↗ 𝑛

𝑛−2 and we arrive at the desired inequality in Theorem 1.8.

Step 4

Suppose the equality in (1.3.3) holds for (𝑋, 𝑔+) as in Thm1.8. Let (𝑋, 𝑀, ˆ𝑔) be its conformal
boundary and ˆ𝑔 = ¯𝑔| 𝑀. If 𝑄( ¯𝑋, 𝑀, [ ¯𝑔]) = 𝑄(B𝑛, S𝑛−1, 𝑑𝑥2), then 𝑌 (𝑀, [ ˆ𝑔]) = 𝑌 (S𝑛−1, [𝑑𝜃2])

from the equality. By Thm1.3, (𝑀, [ ˆ𝑔]) is the round metric on S𝑛−1. Then Thm1.10 implies that

(𝑋, 𝑔+) is the standard hyperbolic space.

Now we suppose 𝑄( ¯𝑋, 𝑀, [ ¯𝑔]) < 𝑄(B𝑛, S𝑛−1, 𝑑𝑥2).

In this case the minimizer for Type II

Yamabe invariant can be realized, say ¯𝑔 = 𝜌2𝑔+. Note that we defined ˜𝜆𝑞 and proved inequality for
𝑛−2 in step 3. This is because we are not sure whether minimizer for 𝑄( ¯𝑋, 𝑀, [ ¯𝑔]) exists.
each 𝑞 < 𝑛

Now since we have got minimizer, we can run the previous method directly and get

∫

2

𝑋

𝑣2−𝑛𝑄dV𝑔+ ≤

𝑉 (𝑀, ˆ𝑔)

𝑛−1
𝑛−3
4(𝑛 − 1)

(cid:104)

𝑄( ¯𝑋, 𝑀, [ ¯𝑔])2 −

4(𝑛 − 1)
𝑛 − 2

𝑌 (𝑀, [ ˆ𝑔])

(cid:105)

19

Given the assumption that equality holds in (1.3.3), it follows that equality also holds in (1.3.6). This

implies 𝑅+ = −𝑛(𝑛 − 1). Combining this with 𝑅𝑖𝑐+ ≥ −(𝑛 − 1)𝑔+, we deduce 𝑅𝑖𝑐+ = −(𝑛 − 1)𝑔+.

We now find ourselves in a situation analogous to that in [CLW17], and their approach is applicable

here as well. For the sake of completeness, we provide a detailed proof.

We also get 𝑄 ≡ 0 and thus ∇2

+𝑣 = Δ+𝑣

𝑣2 𝑔+. Compute ∇2
+𝜌

∇2
+𝑣(𝑋, 𝑌 ) = 𝑔+

(cid:17)

(cid:16)

(∇+)𝑋 ∇+

𝑛 𝑔+. Recall 𝑔 = 𝜌2𝑔+ = 1
1
𝜌
1
𝜌2 ∇+𝜌), 𝑌
(cid:17)

(∇+)𝑋 (

, 𝑌

(cid:16)

(cid:16)

(cid:17)

𝑔+

(∇+)𝑋 ∇+𝜌, 𝑌

+ 2

= −𝑔+

= −

1
𝜌2

(𝑋 𝜌) (𝑌 𝜌)
𝜌3

Taking trace and we get

Δ+𝑣 = −

1
𝜌2 Δ+𝜌 + 2

|∇+𝜌|+
𝜌3

Substitute the above two equations into ∇2

+𝑣 = Δ+𝑣

𝑛 𝑔+ and we have

∇2
+𝜌 =

1
𝑛

(cid:0)Δ+𝜌 −

|∇+𝜌|2
+
𝜌

(cid:1)𝑔+ +

2
𝜌

𝑑𝜌 ⊗ 𝑑𝜌

We now aim to express the preceding equation in terms of 𝑔 = 𝜌2𝑔+. Use (1.2.2), the three equation

above give us

We obtain

Use (1.2.3) and we get

∇2𝜌 =

Δ𝜌
𝑛

𝑔

∇𝑖∇𝑖∇ 𝑗 𝜌 =

1
𝑛

∇ 𝑗 (Δ𝜌)

(1.3.17)

(1.3.18)

𝑅𝑖𝑐+ = 𝑅𝑖𝑐 − (𝑛 − 2) (cid:0)𝜌∇2(

) − 2𝜌2(𝑑

) ⊗ (𝑑

1
𝜌
∇2𝜌 + (cid:0) Δ𝜌
𝜌
|𝑑𝜌|2
𝜌2

1
𝜌
2Δ𝜌
𝑛𝜌

−

1
𝜌

1
𝜌
|𝑑𝜌|2
𝜌2

+ (𝑛 − 1)

(cid:1)𝑔

= 𝑅𝑖𝑐 + (𝑛 − 2)

= 𝑅𝑖𝑐 + (𝑛 − 1) (cid:0)

)(cid:1) − (cid:0)𝜌Δ

1
𝜌

+ (𝑛 − 3) 𝜌2|𝑑 (

1
𝜌

)|2(cid:1)𝑔

(cid:1)𝑔

(1.3.19)

We use (1.3.17) in the last equality. Recall that the scalar curvature of 𝑔 is 0, (1.2.3) implies

𝑅+ = −𝑛(𝑛 − 1) = −𝑛(𝑛 − 1)|∇𝜌|2 + 2(𝑛 − 1) 𝜌Δ𝜌

20

So we have 𝑅𝑖𝑐 ≡ 0. So

∇𝑖∇𝑖∇ 𝑗 𝜌 = ∇ 𝑗 ∇𝑖∇𝑖 𝜌 + 𝑅𝑖𝑐𝑖 𝑗 ∇𝑖 𝜌 = ∇ 𝑗 (Δ𝜌)

(1.3.20)

Compare (1.3.18) and (1.3.20), we have ∇(Δ𝜌) = 0, and therefore Δ𝜌 is constant. Use lemma 1.3.3

and (1.3.13), we get the constant Δ𝜌 = − 𝑛

𝐻 ≠ 0. If not, Δ𝜌 = 0 and 𝜌 = 0 on 𝜕 𝑋, which implies 𝜌 ≡ 0, which is impossible. Set 𝑤 = − 𝑛−1

𝑛−1 𝐻 where 𝐻 is the mean curvature for 𝑔. Apparently
𝑛𝐻 𝜌,

then 𝑤 satisfies

Integrate (Δ𝑤)2

Δ𝑤

𝑤

= 1 in 𝑋

= 0 on 𝑀

(1.3.21)

𝜕𝑤

𝜕𝜈 = 𝑛−1

𝑛𝐻 on 𝑀






𝑛 − 1
𝑛

Vol( ¯𝑋, 𝑔) =

𝑛 − 1
𝑛

∫

𝑋

(Δ𝑤)2dV𝑔

(cid:2)(Δ𝑤)2 − |∇𝑤|2(cid:3)dV𝑔

∫

𝑋
∫

=

=

= (

𝐻

𝑀
𝑛 − 1
𝑛

𝜕𝑤
dS ˆ𝑔
𝜕𝜈
)2 ∫

𝑀

1
𝐻

dS ˆ𝑔

where we used Reilly’s formula in the third line. Therefore we arrive at

∫

𝜕 𝑋

𝑛 − 1
𝐻

dS ˆ𝑔 = 𝑛Vol( ¯𝑋, 𝑔)

Recall that 𝑅𝑖𝑐 = 0. We conclude that (𝑋, 𝑀, 𝑔) is isometric to Euclidean ball by [Mul87].

□

1.4 Geometry on Conformal Boundary Affects Geometry of the Interior

As discussed in the first section, a fundamental principle guiding the research on (asymptot-

ically) Poincaré-Einstein manifolds is to comprehend the intersection between the geometry of

(𝑋, 𝑔+) and the geometry of its conformal boundary. In this section, I will introduce preliminary

results utilized in the preceding section and demonstrate how our Thm1.8 exemplifies this principle.

We start with a toplogy result.

21

Theorem 1.9. Let (𝑋, 𝑔+) asymptotically Poincaré-Einstein manifold and 𝑅𝑖𝑐 ≥ −(𝑛 − 1)𝑔+. If

one connected component of its boundary has non-negative Yamabe invariant, then

𝐻𝑛−1(𝑋; Z) = 0.

In [WY99], E. Witten and S. Yau established the above theorem under the assumption that one

boundary component has a positive Yamabe invariant. They introduced the brane action defined

by

𝐿 (Σ) = Area(Σ) − 𝑛𝑉 (Ω)

(1.4.1)

where Σ = 𝜕Ω, and Ω is a domain in 𝑋. Given the conditions outlined in the theorem, and assuming

a strictly positive Yamabe invariant, they demonstrated the following: 1) 𝐿 (Σ) admits a minimum

through local calculations; 2) there exists a minimum in each nontrivial homology class if the

boundary has a component of positive scalar curvature. Therefore 𝐻𝑛−1(𝑋; Z) = 0.

Later M.Cai and G. Galloway proved the zero Yamabe invaraint case using Riccati equation and

Busemann functions. Let 𝜌 be a boundary defining function and Σ𝜖 = {𝜌(𝑥) = 𝜖 }. They consider

a new Busemann function given by

𝛽𝜖 (𝑥) = 𝑑 (Σ𝜖 , 𝑜) − 𝑑 (𝑥, Σ𝜖 )

𝛽(𝑥) = lim
𝜖→0

𝛽𝜖 (𝑥)

(1.4.2)

Using the Riccati equation, they successfully demonstrated Δ𝛽 ≥ 𝑛 − 1, provided the conformal

boundary has a zero Yamabe invariant. If 𝑋 has more than one end, a carefully chosen ray can be

constructed. Let 𝑏 be the usual Busemann function associated with this ray, resulting in 𝛽 + 𝑏 ≤ 0

with equality at an interior point. Since 𝑅𝑖𝑐 ≥ −(𝑛 − 1)𝑔+, we have Δ𝑏 ≥ −(𝑛 − 1). Now, 𝛽 + 𝑏 is

a subharmonic function with an interior maximum point, implying 𝛽 + 𝑏 ≡ 0. This equality leads

to the splitting (𝑋 = R × Σ, 𝑔+ = 𝑒2𝑟 + ℎ). At 𝑟 = −∞, we encounter a cusp, which contradicts the

asymptotic Poincaré-Einstein condition.

By standard topology argument, we have

Corollary 1.1. Under the same assumption as stated in the preceding theorem, it follows that 𝜕 𝑋

is connected.

22

By this corollary, manifolds in Thm1.8 will have connected boundary and brings us no trouble.

Apart from topology, the conformal geometry of the boundary also affects the metric inside.

We have the following rigidity result

Theorem 1.10. Let (𝑋, 𝑔+) be an asymptotically Poincaré-Einstein manifolds with 𝑅𝑖𝑐 ≥ −(𝑛 −

1)𝑔+. Suppose (𝑋, 𝑔+) has round sphere as its conformal boundary, then (𝑋, 𝑔+) is isometric to

the hyperbolic space (H𝑛, 𝑔H).

The theorem was initially established by Q. Jie in [Qin03] for 𝑛 ≤ 7. In the context of hyperbolic

spaces, we can consider the upper plane model. Q. Jie observed that if (𝑋, 𝑔+) has a round sphere as

its conformal boundary, we can construct coordinate functions and utilize them to apply conformal

transformations, resulting in an uncompact manifold with R𝑛−1 as its boundary, akin to the upper

plane model. Moreover, the scalar curvature is non-negative for the new metric. Consequently,

we can glue two such manifolds along R𝑛−1 to obtain an asymptotically flat manifold ( ˜𝑋, ˜𝑔) with

non-negative scalar curvature. Notably, its Arnowitt-Deser-Misner (ADM) mass 𝑚 𝐴𝐷 𝑀 = 0. The
positive mass theorem [SY79a][SY79b] then implies that ( ˜𝑋, ˜𝑔) is the Euclidean space.

The general case was solved by S.Dutta and M.Javaheri in [DJ10], where they used a totally different

method.

Thm1.8 serves as a compelling illustration of this principle. In the context of our theorem, as

the conformal boundary becomes rounder and rounder, i.e., 𝑌 (𝑀, [𝑔]) ↗ 𝑌 (S𝑛−1, 𝑑𝜃2), the second
inequalities in both (1.2.17) and (1.3.3) compellingly lead to 𝑄( ¯𝑋, 𝑀, [𝑔]) ↗ 𝑄(B𝑛, S𝑛−1, 𝑑𝑥2),

representing the compactification of (H𝑛, 𝑔𝐻). Therefore, our result can be interpreted as follows: as

the conformal boundary approaches the standard sphere, the interior becomes increasingly “close"

to the standard hyperbolic space.

In the context of Thm1.5, where the second inequality in (1.2.17) is strictly satisfied except for the

case of (B𝑛, 𝑑𝑥2), the rigidity theorem can be derived from Thm1.8. The challenge lies in the fact

that we still lack a complete solution to the type II Yamabe problem.

I’d like to mention another result by G.Li, Q.Jie and Y.Shi [LQS14]

23

Theorem 1.11. For any 𝜖 > 0,

𝑛 ≥ 4, there exists 𝛿 > 0 so that for any Poincaré-Einstein

manifold (𝑋, 𝑔+), one gets

|𝐾𝑔+ + 1| ≤ 𝜖

for all sectional curvature 𝐾, provided

𝑌 (𝑀, [ ˆ𝑔]) ≥ (1 − 𝛿)𝑌 (S𝑛−1, [𝑑𝑠2]).

This theorem and Thm1.8 complement each other.

1.5 Some Discussions on Compact Manifolds with Boundary

It is a natural question if the inequality in Theorem 1.8 holds for a compact Riemannian manifold

(𝑀 𝑛, 𝑔) with 𝑅𝑖𝑐 ≥ − (𝑛 − 1) and Π ≥ 1. We are motivated by the observation that some results

for conformally compact manifolds follow from results for compact Riemannian manifolds by a

limiting process. As an illustration, consider the following theorem by Lee.

Theorem 1.12. (Lee [Lee94]) Let (𝑋 𝑛, 𝑔+) be a conformally compact manifold whose conformal in-
finity has nonnegative Yamabe invariant. If 𝑅𝑖𝑐 (𝑔+) ≥ − (𝑛 − 1) 𝑔+ and (𝑋 𝑛, 𝑔+) is asymptotically
Poincare-Einstein, then the bottom of spectrum 𝜆0 (𝑋 𝑛, 𝑔+) = (𝑛 − 1)2 /4.

When the Yambabe invariant of the conformal infinity is positive, Lee’s theorem follows from

the following result for compact Riemannian manifolds.

Theorem 1.13. Let (𝑀 𝑛, 𝑔) be a compact Riemannian manifold with 𝑅𝑖𝑐 ≥ − (𝑛 − 1). If along the

boundary Σ := 𝜕 𝑀 we have the mean curvature 𝐻 ≥ 𝑛 − 1, then the first Dirichlet eigenvalue

𝜆0 (𝑀) ≥

(𝑛 − 1)2
4

.

This theorem has a simple proof. Let 𝑟 be the distance function to Σ. By standard method in

Riemannian geometry, we have

in the support sense. A direct calculation yields

Δ𝑟 ≤ − (𝑛 − 1)

Δ𝑒(𝑛−1)𝑟/2 ≤ −

(𝑛 − 1)2
4

𝑒(𝑛−1)𝑟/2.

24

This implies 𝜆0 (𝑀) ≥ (𝑛−1)2

4

(for technical details see [Wan02]).

We can deduce Lee’s theorem from Theorem 1.13 when the conformal infinity has positive

Yamabe invariant in the following way. As explained in Section 2, we pick a metric ℎ on the

conformal infinity with positive scalar curvature and then we have a good defining function 𝑟 s.t.

near the conformal infinity 𝑔+ has a nice expansion (1.1.2). Then a simple calculation shows that

the mean curvature of the boundary of 𝑋𝜀 := {𝑟 ≥ 𝜀} satisfies

𝐻 = 𝑛 − 1 +

𝑅ℎ
2 (𝑛 − 2)

𝜀2 + 𝑜

(cid:16)

𝜀2(cid:17)

.

As 𝑅ℎ > 0, we have 𝐻 > 𝑛 − 1 if 𝜀 is small enough. By Theorem, 𝜆0 (𝑋𝜀) ≥ (𝑛−1)2
that 𝜆0 (𝑋) ≥ (𝑛−1)2

. It follows
. As the opposite inequality was known by [Maz88], we have 𝜆0 (𝑋) = (𝑛−1)2

4

.

4

4

When the conformal infinity has zero Yamabe invariant, the situation is more subtle. But by an

idea in Cai-Galloway[CG99], a similar argument still works (cf. [Wan02]).

We now come back to Theorem 1.8. By the asymptotic expansion (1.1.2) the second fundamental

form of 𝜕 𝑋𝜀 satisfies

Π+ = (1 + 𝑂 (𝜀)) 𝑔+,

i.e. all the principal curvatures are close to 1. This leads us to consider a compact Riemannian

manifold (𝑀 𝑛, 𝑔) with 𝑅𝑖𝑐 ≥ − (𝑛 − 1) and Π ≥ 1 on its boundary Σ and ask the question whether

the inequality

𝑄 (𝑀, Σ, 𝑔) ≥ 2

√︄

(𝑛 − 1)
(𝑛 − 2)

𝑌 (Σ) if 𝑛 ≥ 4;

(1.5.1)

𝑄 (𝑀, Σ, 𝑔) ≥ 4√︁2𝜋 𝜒 (Σ) if 𝑛 = 3

holds. The answer turns out to be no in general. To construct a counter example, we consider the
(1−|𝑥|2)2 𝑑𝑥2. For 0 < 𝑅 < 1, the

hyperbolic space using the ball model B𝑛 with the metric 𝑔H =

4

Euclidean ball

(cid:40)

𝑥 ∈ B𝑛 : |𝑥|2 =

(cid:41)

𝑥2
𝑖 ≤ 𝑅

𝑛
∑︁

𝑖=1

25

is a geodesic ball in (B𝑛, 𝑔H) and the boundary has 2nd fundamental form Π = 1+𝑅2

2𝑅 𝐼. We now

consider

(cid:40)

𝑀 =

𝑥 ∈ B𝑛 : |𝑥|2 =

𝑖 + 𝑘𝑥2
𝑥2

𝑛 ≤ 𝑅

(cid:41)

,

𝑛−1
∑︁

𝑖=1

where 𝑘 > 0 is close to 1. Then (𝑀, 𝑔H) is a compact hyperbolic manifold with boundary and on

its boundary we have Π ≥ 1 if 𝑘 is sufficiently close to 1 by continuity. Since Σ with the induced

metric is rotationally symmetric, it is conformally equivalent to the standard sphere S𝑛−1. Thus,

𝑌 (Σ) = 𝑌 (cid:0)S𝑛−1(cid:1). But when 𝑘 ≠ 1, the boundary is not umbilic with respect to the Euclidean metric
(cid:16)B𝑛, S𝑛−1(cid:17).

and hence not with respect to 𝑔H either. By [Esc92a] and [Mar07], 𝑄 (𝑀, Σ, 𝑔H) < 𝑄

It follows that the inequality (1.5.1) is false.

Therefore, for a compact Riemannian manifold (𝑀 𝑛, 𝑔) with 𝑅𝑖𝑐 ≥ − (𝑛 − 1) and Π ≥ 1 on

its boundary Σ, it is more subtle to estimate its type II Yamabe invariant in terms of the boundary

geometry. It is an interesting question and we do not have an explicit conjecture. Let us mention

that in a similar setting, namely for a compact (𝑀 𝑛, 𝑔) with 𝑅𝑖𝑐 ≥ 0 and Π ≥ 1 on its boundary

Σ, there is a well-formulated conjecture [Wan19] on the type II Yamabe invariant in terms of the

boundary area.

26

CHAPTER 2

LIOUVILLE TYPE THEOREMS ON MANIFOLDS WITH LOWER CURVATURE
BOUND

One problem that lies in the center of geometric analysis is to understand how geometric conditions,

such as curvature and fundamental forms, exert influence over the solutions of partial differential

equations. In [Wan19], X.Wang proposed a conjecture that for manifolds with boundary, if the

Ricci curvature is nonnegative and second fundament form is positive, then a series of elliptic

PDEs doesn’t admit non-constant solutions. Throughout this chapter, we will always assume that

𝜕 𝑀 = Σ is connected.

2.1 Preparation

X. Wang has posed the following conjecture in [Wan19]:

Conjecture 2.1 (Wang’s conjecture). Let (𝑀, 𝜕 𝑀 = Σ, 𝑔) be a compact Riemannian manifold with

boundary. Suppose 𝑅𝑖𝑐 ≥ 0 on 𝑀, and 𝐼 𝐼 ≥ 1 on Σ where 𝐼 𝐼 is the second fundamental form, then

the following PDE

Δ𝑢 = 0
𝜕𝑢
𝜕𝜈
admits no non-constant positive solution provided 𝜆(𝑞 − 1) ≤ 1 and 𝑞 ≤ 𝑛

= −𝜆𝑢 + 𝑢𝑞 on Σ𝑛−1

on 𝑀 𝑛

(2.1.1)

𝑛−2 unless (𝑀, Σ, 𝑔) is

isometric to (B𝑛, S𝑛−1, 𝑑𝑥2), 𝑞 = 𝑛

𝑛−2 and 𝑢 is given by

𝑢𝑎 =

(cid:104) 2
𝑛 − 2

1 − |𝑎|2
1 + |𝑎|2|𝑥|2 − 2𝑥 · 𝑎

(cid:105) 𝑛−2

2

for some 𝑎 ∈ B𝑛.

The conjecture was proposed for the following reasons. Consider the following functional

𝐸𝑞,𝜆 (𝑢) =

𝑀 |∇𝑢|2dV + 𝜆 ∫
∫
(∫
Σ

𝑢𝑞+1dS)2/(𝑞+1)

Σ

𝑢2dS

𝑠(𝑞, 𝜆) =

inf
𝑢∈𝐻1 (𝑀),𝑢≠0

𝐸 (𝑢)

27

(2.1.2)

For convenience, we drop the index (𝑞, 𝜆) when it brings no confustion. By definition,

𝑠(𝑞, 𝜆) ≤ 𝐸 (1) = 𝜆|Σ|

𝑞−1
1+𝑞

Fix arbitrary 𝑢 and take derivative in the direction of 𝑣, we have

𝜕
𝜕𝑡

𝐸 (𝑢 + 𝑡𝑣)(cid:12)

(cid:12)𝑡=0 =

(∫
Σ

∫

(cid:104)(cid:0)

𝑀

4
𝑞+1

1

𝑢𝑞+1)
∫

2⟨∇𝑢, ∇𝑣⟩ + 2𝜆

∫

∫

𝑢𝑣(cid:1) (

𝑢𝑞+1)

2
𝑞+1

∫

Σ

Σ

𝑣𝑢𝑞(cid:105)

Σ
∫

𝑀

+ 𝑢 − 𝐸 (𝑢) (

𝑢𝑞+1)

1−𝑞
𝑞+1

∫

Σ

𝑣 (cid:0)

𝜕𝑢
𝜕𝜈

𝑢𝑞+1)

1−𝑞

1+𝑞 𝑢𝑞(cid:1)(cid:105)

∫

Σ

𝑣Δ𝑢 +

− 2(

(|∇𝑢|2 + 𝜆𝑢2)) (

=

𝑀

2

(∫
Σ

𝑢𝑞+1)

2
𝑞+1

(cid:104) ∫

𝑀

(2.1.3)

(2.1.4)

Suppose 𝑢 is a critical point of 𝐸, then 𝜕

Δ𝑢 = 0 in 𝑀 and 𝜕𝑢

𝜕𝜈 + 𝑢 − 𝐸 (𝑢)(∫

Σ

𝜕𝑡 𝐸 (𝑢 + 𝑡𝑣)(cid:12)

(cid:12)𝑡=0 = 0 for all smooth 𝑣, and therefore
𝑢𝑞+1)𝑢𝑞 = 0. Since 𝐸 is invariant under scaling, we could

scale 𝑢 to get rid of the coefficients before 𝑢𝑞, yielding (2.1.1). In summary, (2.1.1) arises as the

Euler-Langrangian equation for (2.1.2).

If 𝑞 < 𝑛

𝑛−2, the trace embedding 𝐻1(𝑀) → 𝐿𝑞+1(Σ) is compact (see theorem 6.2 chapter 2
in [Nec11], for example), thereby enabling the attainment of the minimizer denoted as 𝑢𝑞,𝜆. Let
us now consider a fixed value of 𝑞0. As the parameter 𝜆 decreases, the weight of ∫
𝑀 |∇𝑢|2dV
becomes increasingly prominent. To ensure that 𝑢𝑞0,𝜆 attains minimizer, a concomitant decrease in
∫
𝑀 |𝑢𝑞0,𝜆|2dV is expected. For example, we have the following lemma:

Lemma 2.1.1. Let (𝑀, Σ, 𝑔) be a compact Riemannian manifold with boundary. Suppose 𝑠(𝑞, 𝜆) is

achieved by constants for some 𝑞 ≤ 𝑛

𝑛−2. Then for any 𝜇 < 𝜆, 𝑠(𝑞, 𝜆) is only achieved by constants.

Proof. For any fixed 𝑢 ∈ 𝐻1(𝑀) and 𝑢 ≠ 0, 𝐸𝑞,𝜇 (𝑢) is linear function in 𝜇, and therefore concave.

Since 𝑠(𝑞, 𝜇) is the infimum of concave functions, 𝑠(𝑞, 𝜇) is also a concave function in 𝜇. Suppose

𝑠(𝑞, 𝜆) is achieved by constant, we have 𝑠(𝑞, 𝜆) = 𝜆|Σ|

𝑞−1
𝑞+1 . We also have 𝑠(𝑞, 0) = 0. By concavity,

𝑠𝑞,𝜇 ≥ 𝜇|Σ|

𝑞−1
𝑞+1

for 𝜇 < 𝜆. At the same time, we have 𝑠𝑞,𝜇 ≤ 𝐸𝑞,𝜇 (1) = 𝜇|Σ|

1−𝑞
1+𝑞 . So 𝑠(𝑞, 𝜇) is achieved by

constants.

28

Suppose 𝑠(𝑞, 𝜇) is achieved by some non-constant 𝑢 and 𝜇 < 𝜆, i.e. 𝐸𝑞,𝜇 (𝑢) = 𝐸𝑞,𝜇 (1) = 𝑠(𝑞, 𝜇).
Since 𝑢 is non-constant, we must have ∫

𝑞−1
𝑞+1 . Use 𝑢

∫
Σ

𝑀 |∇𝑢|2 > 0 and therefore

< |Σ|

𝑢2
𝑢 (𝑞+1) )2/(𝑞+1)

(∫
Σ

as the test function for (𝑞, 𝜆), we obtain

𝐸𝑞,𝜆 (𝑢) =

=

Σ
𝑢𝑞+1)2/(𝑞+1)

𝑀 |∇𝑢|2 + 𝜆 ∫
∫
(∫
Σ
𝑀 |∇𝑢|2 + 𝜇 ∫
∫
(∫
Σ

Σ
𝑢𝑞+1)2/(𝑞+1)

𝑢2

𝑢2

+ (𝜆 − 𝜇)

∫
Σ

𝑢2

𝑢𝑞+1)2/(𝑞+1)

(∫
Σ

<𝜇|Σ|

𝑞−1
𝑞+1 + (𝜆 − 𝜇)|Σ|

𝑞−1
𝑞+1 = 𝑠(𝑞, 𝜆)

which is contradiction since we assumer 𝑠(𝑞, 𝜆) is achieved by constant.

□

Note that 𝑢𝑞,0 and 𝑢1,𝜆 are both constants. Therefore, for values of 1 < 𝑞 < 𝑛

𝑛−2, an intriguing
possibility emerges: for each value of 𝑞, there might exist a threshold 𝜆𝑞 such that when 𝜆 < 𝜆𝑞,

the minimizer 𝑢𝜆,𝑞 will take constant values.

This phenomena was first found as Beckner inequality [Bec93]:

Theorem 2.1 (Beckner’s inequality). For unit disk with Euclidean metric and 𝑦 ∈ 𝐻1(B𝑛), we have

𝑞−1
𝑞+1

𝑛−1(

𝑐

∫

S𝑛−1

𝑢𝑞+1dS)

2
𝑞+1 ≤

1
𝜆

∫

B𝑛

|∇𝑢|2dV +

∫

S𝑛−1

𝑢2dS

(2.1.5)

provided that 1 ≤ 𝑞 ≤ 𝑛

𝑛−2 and 𝜆(𝑞 − 1) ≤ 1, where 𝑐𝑛−1 = 2𝜋(𝑛−1)/2/Γ (cid:0)(𝑛 − 1)/2(cid:1) is the volume

of 𝑛 − 1 round sphere.

It follows from the inequality that the minimizers of 𝐸𝜆,𝑞 for unit ball are exclusively realized by

constant functions. This intriguingly gives rise to the conjecture that (2.1.1) admits no non-constant

solutions.

When 𝜆(𝑞 − 1) > 1, however, the 𝑢 ≡ 1 is no longer minimizer for unit balls. The second

variation of 𝐸𝜆,𝑞 at 𝑢 ≡ 1 in the direction of 𝑣 is

𝜕2
𝜕𝑡2

𝐸𝜆,𝑞 (1 + 𝑡𝑣)(cid:12)

(cid:12)𝑡=0 = −

2
|Σ|2/(𝑞+1)

∫

(cid:104)

−

∫

(cid:0)

𝑣Δ𝑣 +

Σ

𝑀
|∇𝑣|2 − 𝜆(𝑞 − 1)

𝜕𝑣
+ 𝜆𝑣(cid:1)𝑣 − 𝜆𝑞𝑣2(cid:105)
𝜕𝜈
∫

2
|Σ|2/(𝑞+1)
We can pick 𝑣 to be the function associated to the first Steklov eigenvalue.

𝑣2(cid:105)

=

𝑀

Σ

(cid:104) ∫

(2.1.6)

29

Definition 2.1.1. The first Steklov eigenvalue is

𝜆 =

inf
𝑢∈𝐻1 (𝑀),𝑢≠0

∫
𝑀 |∇𝑢|2
∫
𝑢2
Σ

The corresponding Euler-Lagrangian equation, or the eigenfunction equation, is

Δ𝑢 = 0 in 𝑀
𝜕𝑢
𝜕𝜈

= 𝜆𝑢 on Σ

(2.1.7)

(2.1.8)

It’s well known that the first Steklov eigenvalue for unit ball is 1 with 𝑛 eigenfunctions given by

coordinate functions. Pick 𝑣 to be Steklov eigenfunction in (2.1.6), and we have

𝜕2
𝜕𝑡2

𝐸𝜆,𝑞 (1 + 𝑡𝑣)(cid:12)

(cid:12)𝑡=0 =

2(1 − 𝜆(𝑞 − 1))
𝑐2/(𝑞+1)
𝑛−1

∫

S𝑛−1

𝑣2

As a consequence, the minimization of 𝐸𝜆,𝑞 by 𝑢 ≡ 1 is unsuccessful if 𝜆(𝑞 − 1) > 1. However,

the minimizer exist since the trace embedding is compact. This implies that the minimizer is a

non-constant solution of (2.1.1). Therefore, the condition 𝜆(𝑞 − 1) ≤ 1 is crucial and cannot be

improved. These insights serve to clarify the conjecture for B𝑛.

It’s noteworthy to mention that the conjecture is fully resolved for unit balls as demonstrated in

[GL23]. A natural progression from here is to delve into the intricate connection between geometric

properties and the behavior of solutions of (2.1.1). This exploration is driven by the question of

how geometric attributes influence the solutions of PDEs. A similar problem was resolved:

Theorem 2.2 (B.Véron and L.Véron [BV91]). Let (𝑀, Σ, 𝑔) be a compact Riemannian manifold

with boundary.

−Δ𝑢 + 𝜆𝑢 = 𝑢𝑞
𝜕𝑢
𝜕𝜈

= 0

on 𝑀 𝑛

on Σ𝑛−1

admits no non-constant solutions provided that 𝜆 > 0, 1 < 𝑞 < 𝑛+2

𝑛−2 and 𝑅𝑖𝑐 ≥ (𝑛−1)(𝑞−1)𝜆

𝑛

In the context of Wang’s conjecture, there is famous Escobar conjecture.

30

(2.1.9)

𝑔.

Conjecture 2.2 (Escobar conjecture). Let (𝑀, Σ, 𝑔) be a compact Riemannian manifold with

boundary. Suppose 𝑅𝑖𝑐 ≥ 0 on 𝑀, and 𝐼 𝐼 ≥ 1 on Σ. Then

∫

𝑀

|∇𝑢|2 ≥

∫

Σ

𝑢2

(2.1.10)

i.e. the first Steklov eigenvalue is no less than 1.

Under the condition of non-negative sectional curvature Escobar’s conjecture was completely

solved by C.Xia and C.Xiong in ([XX19]).

The insights gleaned from these findings, in conjunction with the outcomes concerning B𝑛, culmi-

nate in Wang’s conjecture.

If this conjecture holds true, it would lead to fascinating geometric implications. For instance,

an intriguing consequence would be an upper bound on the area of Σ.

Conjecture 2.3. Let (𝑀, Σ, 𝑔) be as in conjecture 2.1. Then

𝐴𝑟𝑒𝑎(Σ) ≤ 𝐴𝑟𝑒𝑎(S𝑛−1)

(2.1.11)

Moreover, this inequality would only be realized by unit spheres as the boundary of unit disks.

. We can view this as an extension of the Bishop volume comparison theorem.

Consider the 𝐸𝑞, 1

𝑞−1

. For 𝑞 < 𝑛

𝑛−2 , 𝑠(𝑞, 1

𝑞−1) can be achieved for some smooth function.

If

conjecture 2.1 holds true, then the only possible minimizer are constants, which implies

(𝑞 − 1) ∫
(∫
Σ

𝑀 |∇𝑢|2 + ∫
Σ
𝑢(𝑞+1))2/(𝑞+1)

𝑢2

𝑞−1
𝑞+1

≥ |Σ|

for any 𝑢 ∈ 𝐻1(𝑀) and 𝑢 ≠ 0. Let 𝑞 ↘ in above inequality and we get

𝑀 |∇𝑢|2 + ∫
∫
2
𝑛−2
𝑢2(𝑛−1)/(𝑛−2)) (𝑛−2)/(𝑛−1)

𝑢2

Σ

(∫
Σ

≥ |Σ|

1
𝑛−1

(2.1.12)

Recall the definition for type II Yamabe invariant in (1.2.14). Since 𝑅𝑖𝑐 ≥ 0 and 𝐼 𝐼 ≥ 𝑔|Σ, we get

𝑄(𝑀, Σ, 𝑔) ≥

4(𝑛−1)
𝑛−2 𝑠( 𝑛

𝑛−2, 𝑛−2

2 ). By the second inequality in (1.2.17), we finally arrive at (2.1.11).

31

The relationship between curvature and volume has a long and storied history in geometry.

For manifolds without boundaries, the Bishop comparison theorem is a pivotal result. It asserts

that a lower bound on the Ricci curvature results in an upper bound on the volume of geodesic

balls. In the context of manifolds with boundaries, where we encounter second fundamental forms,

a lower bound for this form could imply an upper bound for the distance from the boundary to

the interior. When coupled with the Bishop comparison method, it naturally leads us to surmise

such upper bounds on volume and area. However, tackling this problem is notably challenging.

The conjecture we’ve presented offers a promising avenue to approach and potentially solve this

intriguing problem.

A lot of work has been invested in exploring this conjecture; however, a significant portion of

its components remain unsolved. As I see it, there are two primary challenges that contribute to

the difficulty of addressing this conjecture.

Firstly, there exists a lack of comprehensive understanding regarding how Ricci curvature influ-

ences the solutions of (2.1.1). A notable advance in this direction was made in [GHW19], where it

was demonstrated that by strengthening the Ricci curvature assumption to 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒 ≥

0, a beneficial weight function emerges. This weight function proves advantageous during inte-

gration by parts, effectively nullifying bothersome boundary terms. This paves the way for the

derivation of a weighted version of Reilly’s formula, leading to partial results. Unfortunately, this

method falters when the assumption is relaxed to non-negative Ricci curvature. In fact, no results

exist in this setting.

I have obtained results in general Riemannian manifolds using different techniques. However,

it’s currently unclear how Ricci curvature, or even sectional curvature, influences the estimation in

my approaches. I’m working to get over these difficulties to get a uniform estimate under curvature

assumptions.

The second primary challenge emerges when 𝑞 ↗ 𝑛

𝑛−2, particularly when the equality is
(Σ) ↩→ 𝐻1(𝑀) becomes merely continuous

reached. Notably, at 𝑞 = 𝑛

𝑛−2, the embedding 𝐿 2(𝑛−1)

𝑛−2

32

without the compactness property, raising uncertainties about the existence of the minimizer. It’s

worth emphasizing that in this instance, (2.1.2) takes on a form reminiscent of the second-type

Yamabe invariant defined in the preceding section. These intricacies contribute to the heightened

complexity of addressing this conjecture.

Given this challenge, it might be worthwhile to concentrate on cases where 𝑞 is close to 1.

When we take the derivative with respect to 𝑝 at 𝑝 = 1, we obtain a log-Sobolev inequality. The

verification of the log-Sobolev inequality would provide confidence in Wang’s conjecture.

I’m

working in this direction and partial results are obtained. The results mentioned above will be

presented in the following sections.

2.2 Pseudo Differential Operator

We start with an exploration of Dirichlet-to-Neumann operator. Using the property of Dirichlet-

to-Neumann operator we can derive a non-existence theorem for (2.1.1) on general manifold with

boundary without adding any restriction for curvature. In this section all the integration and Sobolev

spaces will be with respect to Σ unless stated otherwise.

Definition 2.2.1.

𝐷𝑁 : 𝐻1(Σ) −→ 𝐿2(Σ)

𝐷𝑁 ( 𝑓 ) =

𝜕𝑢
𝜕𝜈

(2.2.1)

where 𝑢 is the harmonic extension for 𝑓 , i.e.

Δ𝑢 = 0

in 𝑀

𝑢|Σ = 𝑓

on Σ

It’s well known that 𝐷𝑁 is a first order elliptic pseudo differential operator. See [Tay96] chapter

1, for example. As a result

𝐶1∥∇ 𝑓 ∥ 𝐿2 ≤ ∥𝐷𝑁 ( 𝑓 ) ∥ 𝐿2 ≤ 𝐶2∥∇ 𝑓 ∥ 𝐿2

(2.2.2)

Define ˜𝐿2(Σ) := { 𝑓 ∈ 𝐿2(Σ) : ∫
Σ
have ∥ 𝑓 ∥𝑙2 ≤ 𝐶 ∥∇ 𝑓 ∥ 𝐿2 for 𝑓 ∈ ˜𝐻1(Σ). Then (2.2.2) can be rewritten as

𝑓 = 0} and ˜𝐻1(Σ) = 𝐻1(Σ) ∩ 𝐴. Using Poincaré lemma, we

𝐶1∥ 𝑓 ∥𝐻1 ≤ ∥𝐷𝑁 ( 𝑓 ) ∥ 𝐿2 ≤ 𝐶2∥ 𝑓 ∥𝐻1

(2.2.3)

33

for 𝑓 ∈ ˜𝐻1(Σ).

Suppose 𝑓 ∈ 𝐻1(Σ) and 𝐷𝑁 ( 𝑓 ) = 0. By (2.2.2) and employing the standard bootstrapping

strategy, 𝑓 must be a smooth function. Then, by the definition of 𝐷𝑁 and the maximal principle,

the harmonic extension of 𝑓 remains constant, implying 𝑓 itself is constant. If we restrict to ˜𝐻1(Σ),
then 𝑓 must identically vanish. Consequently, 𝐷𝑁 is injective when restricted to ˜𝐻1(Σ).

Further more, 𝐷𝑁 is self-adjoint on 𝐶∞(Σ). Let 𝑓 , 𝑔 ∈ 𝐶∞(Σ) and 𝑢, 𝑣 be their harmonic extension

to 𝑀 respectively. Then

∫

Σ

𝐷𝑁 ( 𝑓 )𝑔 =

=

=

∫

Σ
∫

𝑀

∫

Σ

𝜕𝑢
𝜕𝜈

𝑔

⟨∇𝑢, ∇𝑣⟩

𝜕𝑣
𝜕𝜈

𝑓 =

∫

Σ

𝑓 𝐷𝑁 (𝑔)

(2.2.4)

Let me introduce theorem 5.5 in chapter 3 of [LM90].

Theorem 2.3. Let 𝐸 be a hermitian vector bundle with connection over a compact Riemannian

manifold, Γ(𝐸) the smooth sections for 𝐸. Suppose 𝑃 : Γ(𝐸) → Γ(𝐸) is elliptic and self-adjoint,

then there is an 𝐿2-orthogonal direct sum decomposition:

Γ(𝐸) = 𝑘𝑒𝑟 (𝑃) ⊕ 𝐼𝑚(𝑃)

The statement is for hermitian bundles, but the argument works for real bundles with inner product

structure as well. We already found the kernel of 𝐷𝑁 is given by constant R. Given 𝑔 ∈ 𝐶∞(Σ),
𝑔, which lies in ˜𝐿2(Σ). By the theorem above, 𝐷𝑁 is

its 𝐿2-orthogonal projection to R is 𝑔 − 1
Σ
surjective from 𝐶∞(Σ) ∩ ˜𝐿2(Σ) → 𝐶∞(Σ) ∩ ˜𝐿2(Σ), and therefore bijective.
Now assume 𝑔 ∈ ˜𝐿2(Σ) which doesn’t have to be smooth, then there exists a sequence {𝑔𝑖} ∈ 𝐶∞(Σ)
so that 𝑔𝑖 → 𝑔 in 𝐿2(Σ). We can further assume that ∫
Σ

𝑔𝑖 = 0 by taking 𝑔𝑖 − 1
|Σ|

𝑔𝑖 instead. Then

∫
Σ

∫
Σ

there exists 𝑓𝑖 so that 𝐷𝑁 ( 𝑓𝑖) = 𝑔𝑖. Use the first inequality in (2.2.3), { 𝑓𝑖} is a Cauchy sequence and

therefore converges to some 𝑓 ∈ 𝐻1(Σ). It easy to see that 𝐷𝑁 ( 𝑓 ) = 𝑔, and thus 𝐷𝑁 is actually

surjective. Combine everything above, and we arrive at

34

Lemma 2.2.1. When viewed as a map from 𝐻1(Σ) to 𝐿2(Σ), the image of 𝐷𝑁 is ˜𝐿2(Σ), and its

kernel is R. Furthermore, 𝐷𝑁 is self-adjoint when restricted to 𝐶∞(Σ).

For the Laplace equation, the existence of Green’s function is a key tool for solving the equation.

Given a compact manifold Σ without boundary, there exists a unique function 𝐺 (𝑥, 𝑦) satisfying
Δ𝑦𝐺 (𝑥, 𝑦) = 𝛿𝑥 (𝑦). Consequently for any 𝑓 ∈ 𝐿2(Σ) and ∫
Σ

𝑓 = 0, 𝑢(𝑥) := ∫
Σ

𝐺 (𝑥, 𝑦) 𝑓 (𝑦)𝑑𝑦

solves Δ𝑢 = 𝑓 . In the context of Dirichlet-to-Neumann operator, a comparable kernel is anticipated.
For 𝑢 ∈ 𝐶∞(Σ) ∩ ˜𝐿2, define

𝑇 (𝑢) = 𝐷𝑁 −1(𝑢 −

1

∫

|Σ|

Σ

𝑢)

(2.2.5)

It defines a

𝑇 is well-defined by lemma 2.2.1, and 𝑇 (𝑢) is also 𝐶∞(Σ), and thus in D (Σ).
bilinear form 𝐵(𝑢, 𝑣) = ∫
Σ

𝑇 (𝑢)𝑣. Obviously 𝐵 satisfies the conditions in theorem 2.2.15 (explicit

formulation will be given at the end of this section), and therefore there exists a kernel 𝐾 ∈ D (Σ×Σ)

such that

𝐵(𝑢, 𝑣) =

∫

Σ

𝑇 (𝑢)(𝑥)𝑣(𝑥)𝑑𝑥 =

∫

Σ×Σ

𝑢(𝑥)𝑣(𝑦)𝐾 (𝑥, 𝑦)𝑑𝑥𝑑𝑦

for any 𝑢, 𝑣 ∈ 𝐶∞(Σ). Since 𝐷𝑁 is a first order elliptic pseudo-differential operator, 𝑇 is elliptic of

order −1, and we have the following estimate for 𝐾 from chapter 1, section 2 in [Tay96]

Lemma 2.2.2. 𝐾 is 𝐶∞ off the diagonal in Σ × Σ, and

|𝐾 | ≤ 𝐶𝑑 (𝑥, 𝑦)2−𝑛

(2.2.6)

Σ has dimention 𝑛 − 1, and therefore ∫

Σ×Σ

|𝑢(𝑥)𝑣(𝑦)𝐾 (𝑥, 𝑦)|𝑑𝑥𝑑𝑦 < ∞. Therefore we could

apply Fubini theorem

∫

Σ

𝑇 (𝑢)(𝑥)𝑣(𝑥)𝑑𝑥 =

∫

Σ

𝑣(𝑦) (cid:0)

∫

Σ

𝑢(𝑥)𝐾 (𝑥, 𝑦)𝑑𝑥(cid:1) 𝑑𝑦

Fix 𝑢 and view 𝑣 as the test function, we have

𝑇𝑢(𝑥) =

∫

Σ

𝐾 (𝑥, 𝑦)𝑢(𝑦)𝑑𝑦

35

i.e.

𝑢 −

1

∫

|Σ|

Σ

𝑢 =

∫

Σ

𝐾 (𝑥, 𝑦)𝐷𝑁 (𝑢) (𝑦)𝑑𝑦

(2.2.7)

Next we are going to use this expression to prove a non-existence theorem. For convenience, we

scale 𝑢 so that ∫
Σ

𝑢𝑞+1 = 1, and (2.1.1) becomes

on 𝑀 𝑛

= −𝜆𝑢 + 𝑠(𝑞, 𝜆)𝑢𝑞 on Σ𝑛−1

(2.2.8)

Δ𝑢 = 0
𝜕𝑢
𝜕𝜈
𝑢𝑞+1 = 1

∫

Σ

Use (2.2.7) for (2.2.8),

(cid:0)𝑢 −

1

∫

Σ

Σ

𝑢(𝑥)(cid:1) =

∫

Σ

𝐾 (𝑥, 𝑦) (−𝜆𝑢 + 𝑠𝑢𝑞) (𝑦)𝑑𝑦

(2.2.9)

By (2.2.6), 𝐾 (𝑥, ·) is 𝐿 𝑝 for any 𝑝 < 𝑛−1

𝑛−2. Since Σ is compact, we can find a 𝐶 independant of 𝑥

such that

∥𝐾 (𝑥, ·)∥ 𝑝 ≤ 𝐶, ∀𝑥 ∈ Σ

In the remaining of this section, 𝐶 is a constant that depends on metric and 𝑞, but not 𝜆. Let

𝑝∗ =

𝑝
𝑝−1 be the conjugate of 𝑝, and we have 𝑝∗ > 𝑛 − 1. Then by Hölder inequality, the left hand

side of (2.2.9) can be bounded by

𝐿𝐻𝑆 ≤ 𝐶 ∥ − 𝜆𝑢 + 𝑠𝑢𝑞 ∥ 𝑝∗

≤ 𝐶𝜆(∥𝑢∥ 𝑝∗ + ∥𝑢𝑞 ∥ 𝑝∗)

(2.1.3) is used for the second inequality. Let 𝑀 = sup 𝑢, and 0 < 𝑡 < 1.

𝐿𝐻𝑆 ≤ 𝐶𝜆𝑀 𝑡 (∥𝑢1−𝑡 ∥ 𝑝∗ + ∥𝑢𝑞−𝑡 ∥ 𝑝∗)

(2.2.10)

We aim to bound the right-hand side by ∫
Σ

𝑢𝑞+1 = |Σ|. By Hölder’s inequality, this is achievable

when (𝑞 − 𝑡) 𝑝∗ ≤ 𝑞 + 1, which is equivalent to 𝑞 ≤

𝑡 𝑝∗+1
𝑝∗−1 . Given that 𝑞 < 𝑛

𝑛−2, we can choose

0 < 𝑡 < 1 and 𝑝∗ > 𝑛 − 1 to meet the requirement.

36

For the left hand side, we might take 𝑥 to be the maximal point for 𝑢. Again, by Hölder inequality

and ∫
Σ

𝑢𝑞+1 = |Σ|, we have 1
|Σ|

∫
Σ

𝑢 ≤ 1. Together, we have

𝑀 − 1 ≤ 𝐶𝜆𝑀 𝑡

Since 𝑡 < 1, we arrive at the following

Lemma 2.2.3. Let 𝑢 be a solution of (2.2.8). Then we have

∥𝑢∥∞ ≤ 𝐶 (𝑀, 𝑛, 𝑞)

(2.2.11)

provided that 𝜆 < 1, where 𝑁 is a constant that depends on 𝑛, 𝑞 and 𝐶 (𝑀, 𝑛, 𝑞) depends on the

𝑛, 𝑞 and Riemannian manifold 𝑀.

Based on this 𝐿∞ estimate, we can prove a non-existence theorem:

Theorem 2.4. For each 1 < 𝑞 < 𝑛

𝑛−2, there exists 𝜆𝑞 so that (2.2.8) only admits constant solutions

for 𝜆 ≤ 𝜆0. As a consequence, for these 𝜆’s

𝑠𝜆,𝑞 = 𝜆𝐴(Σ)

𝑞−1
𝑞+1

We start with a lemma

Lemma 2.2.4. Let 𝑢 be a harmonic function on (𝑀, Σ, 𝑔), then

where 𝜇 is the first Steklov eigenvalue.

∫

𝑀

|∇𝑢|2 ≤ 𝜇

∫

Σ

𝜕𝑢
𝜕𝑛

|

|2

(2.2.12)

Proof. Note that the inequality above is invariant under translation, so it suffices to prove the case
∫
𝑀

𝑢 = 0.

∫

𝑀

|∇𝑢|2 =

≤

≤

∫

𝑢

𝜕𝑢
𝜕𝑛

Σ
∫
𝜖
2
𝜖
2𝜇

Σ
∫

𝑢2 +

1
2𝜖

|∇𝑢|2 +

∫

|

Σ

|2

𝜕𝑢
𝜕𝑛
∫

1
2𝜖

𝜕𝑢
𝜕𝑛

|

|2

Σ

𝑀

37

Therefore

(2𝜖 −

𝜖 2
𝜇

)

∫

𝑀

|∇𝑢|2 ≤

∫

Σ

𝜕𝑢
𝜕𝑛

|

|2

The infimum of the quadratic on the left hand side is achieved for 𝜖 = 𝜇 and, and the lemma

follows.

Proof of the theorem

∫

𝜇

𝑀

|∇𝑢|2 ≤

=

=

□

∫

Σ

∫

Σ
∫

𝜕𝑢
𝜕𝑛

|

|2

(𝑠𝑢𝑞 − 𝜆𝑢)

𝜕𝑢
𝜕𝑛

(𝑠𝑞𝑢𝑞−1 − 1)|∇𝑢|2

(2.2.13)

𝑀
∫

≤ 𝜆

(𝑞𝑢𝑞−1 − 1)|∇𝑢|2

≤ 𝐶𝜆

𝑀
∫

𝑀

|∇𝑢|2

Therefore, if 𝜆 is small, 2.1.2 admit no non-constant minimizer.

□

In Theorem 2.4, we investigated the solutions of (2.2.8), which arises as the minimizer for the

functional. The same method can be applied to examine solutions for (2.1.1), which are not

necessarily minimizers, but the trade-off is that 𝑞 < 𝑛−1
𝑛−2.

Theorem 2.5. For each 1 < 𝑞 < 𝑛−1

𝑛−2, there exists 𝜆𝑞 so that the equation only admits constant

solutions for 𝜆 ≤ 𝜆0.

Proof. The method is similar, and I will only show the different parts. Recall that for equation
(2.2.8) we have ∫
Σ

𝑢𝑞+1 = |Σ| and we can control the right hand side of estimate (2.2.10). But for

(2.1.1) we have to derive such an estimate. Integrate (2.1.1) by parts and we have

By Hölder inequality

0 =

∫

𝑀

Δ𝑢 =

∫

Σ

−𝜆𝑢 + 𝑢𝑞+1

∫

𝑢𝑞 = 𝜆

∫

𝑢 ≤ 𝜆(

𝑢𝑞)

1

𝑞 |S𝑛−1|

𝑞−1
𝑞

𝑢𝑞 ≤ 𝜆

𝑞

𝑞−1 |S𝑛−1|

38

∫

∫

⇒

(2.2.14)

Now run the method for the previous theorem, the estimate (2.2.10) becomes

𝐿𝐻𝑆 ≤ 𝐶 (𝜆 + 1) (∥𝑢1−𝑡 ∥ 𝑝∗ + ∥𝑢𝑞−𝑡 ∥ 𝑝∗)

where 0 < 𝑡 < 1 and 𝑝∗ > 𝑛 − 1. We aim to bound the right-hand side by ∫
Σ
𝑡 𝑝∗
𝑝∗−1, which implies 𝑞 < 𝑛−1
𝑛−2.

(𝑞 − 𝑡) 𝑝∗ ≤ 𝑞. This is feasible if 𝑞 ≤

𝑢𝑞, which requires

□

Remark 2.2.1. In theorem 2.4, we initiate with an 𝐿𝑞+1 bound, while in theorem 2.5, we can only
derive an 𝐿𝑞 estimate. This is the rationale behind assuming 𝑞 < 𝑛−1

𝑛−2 instead of 𝑞 < 𝑛
𝑛−2.

To end this section, I will give explicit formulation of Schwartz kernel theorem. Let 𝑀 be two

compact Riemannian manifolds. We can define the following seminorms on 𝐶∞(𝑀) by

|𝑢|𝑘 := sup
𝑥∈𝑀

∑︁

𝛼≤𝑘

|∇𝛼𝑢(𝑥)|

These seminorms give topology to 𝐶∞(𝑀). A linear map 𝑇 from 𝐶∞(𝑀) to R is continuous

provided that there exists some 𝐶 and 𝑘

for all 𝑢 ∈ 𝐶∞(𝑀). Let D denote the space of distribution on 𝑀, i.e. all the continuous maps in

𝑇 (𝑢) ≤ 𝐶 |𝑢|𝑘

the sense as above.

Suppose there is another Riammnian manifold 𝑁, and a map

𝑇 : 𝐶∞(𝑀) → D (𝑁)

𝑇𝑢 is a continuous operator on 𝐶∞(𝑁), and thus giving rise to a bilinear form 𝐵 by the following:

𝐵 : 𝐶∞(𝑀) × 𝐶∞(𝑁) → R

𝐵(𝑢, 𝑣) = ⟨𝑇𝑢, 𝑣⟩,

𝑢 ∈ 𝐶∞(𝑀), 𝑣 ∈ 𝐶∞(𝑁)

Finally, define 𝑢 ⊗ 𝑣 ∈ 𝐶∞(𝑀 × 𝑁) by

𝑢 ⊗ 𝑣(𝑥, 𝑦) := 𝑢(𝑥)𝑣(𝑦), 𝑥 ∈ 𝑀, 𝑦 ∈ 𝑁

Given all these preparations, we have the Schwartz kernel theorem

39

Theorem 2.6 (Schwartz kernel theorem). For any 𝐵 as in above, there exists a distribution 𝐾 ∈

D (𝑀 × 𝑁) so that for 𝑢 ∈ 𝐶∞(𝑀) and 𝑣 ∈ 𝐶∞(𝑁) we have

𝐵(𝑢, 𝑣) = ⟨𝑢 ⊗ 𝑣, 𝐾⟩

(2.2.15)

2.3 Bootstrapping Strategy

The 𝐿∞ estimate (2.2.11) can also be derived using standard bootstrapping strategy, and it’s

more straightforward. In this section (𝑞, 𝜆) will dropped for 𝑠(𝑞, 𝜆) and 𝐸𝑞,𝜆 (𝑢) for simplicity.

𝐶 will be a constant that does not depend on 𝜆 or 𝑞 and might change from line to line. All the

integral and norms will be on the boundary Σ.

We start from 𝑥0 = 𝑞 + 1, and choose 𝑥𝑘 inductively by

If 𝑢 ∈ 𝐿𝑥𝑘 for 𝑥𝑘 ≥ 𝑞 + 1, then 𝜕𝑢

−

1
𝑛 − 1

(2.3.1)

𝑥𝑘
𝑞 . By Hölder inequality we have

=

1
𝑞
𝑥𝑘+1
𝑥𝑘
𝜕𝑛 = −𝜆𝑢 + 𝑠𝑢𝑞 ∈ 𝐿
𝜕𝑢
𝜕𝑛

≤ 𝜆∥𝑢∥ 𝑥𝑘

∥ 𝑥𝑘
𝑞

𝑞

∥

+ 𝑠∥𝑢𝑞 ∥ 𝑥𝑘

𝑞

≤ 𝜆|Σ|

𝑞
𝑥𝑘

− 1
𝑥𝑘 ∥𝑢∥𝑥𝑘 + 𝑠∥𝑢∥

𝑞
𝑥𝑘

≤ 𝐶𝜆(∥𝑢∥𝑥𝑘 + ∥𝑢∥

𝑞
𝑥𝑘 )

We used (2.1.3) in the third line. Since 𝑞 ≤ 𝑛
𝑞
𝑥𝑘

𝑛−2, 𝑥𝑘 is increasing, and 𝑥𝑘 ≥ 𝑞 + 1. Therefore,
is bounded from both below and above, and that’s why the constant 𝐶 in the third line can

− 1
𝑥𝑘

be made independent of 𝑞. Since Dirichlet-to-Neumann operator is elliptic of order 1 (see chapter

1 of [Tay96], for example), we have

∥𝑢∥ 𝑥𝑘

𝑞 ,1 ≤ 𝐶 (∥

𝜕𝑢
𝜕𝑛

∥ 𝑥𝑘
𝑞

+ ∥𝑢∥ 𝑥𝑘

𝑞

)

≤ 𝐶 (𝜆 + 1) (∥𝑢∥𝑥𝑘 + ∥𝑢∥

𝑞
𝑥𝑘 )

By Sobolev embedding theorem on the boundary and our choice of 𝑥𝑘 , we hav

∥𝑢∥𝑥𝑘+1 ≤ 𝐶 ∥𝑢∥𝑥𝑘/𝑞,1 ≤ 𝐶 (cid:0)∥𝑢∥𝑥𝑘 + ∥𝑢∥

(cid:1)

𝑞
𝑥𝑘

(2.3.2)

We used the assumption that 𝜆 < 1 in the second inequality. Note that the constant 𝐶 might change

for different 𝑘. But we are only taking finite bootstripe steps, so we can pick a universal constant 𝐶.

40

Lemma 2.3.1. The sequence 𝑥𝑘 will be negative in 𝐾 (𝑛, 𝑞) steps, and 𝐾 (𝑛, 𝑞) depends only on the

dimension and an upper bound for 𝑞.

This will imply an 𝐿∞ bound by Sobolev inequality.

Proof of lemma:

Let 𝑦𝑘 = 1
𝑥𝑘

and rewrite 2.3.1 as

𝑦𝑘+1 − 𝑦𝑘 = (𝑞 − 1)𝑦𝑘 −

1
𝑛 − 1

(2.3.3)

which implies 𝑦𝑘+1 < 𝑦𝑘 if 𝑦𝑘 <

the inequality. So by induction 𝑦𝑘 <

1
(𝑞−1)(𝑛−1)
1
(𝑞−1)(𝑛−1)

. By our assumption that 𝑞 < 𝑛

𝑛−2 we see that 𝑦0 satisfy
and 𝑦𝑘+1 < 𝑦𝑘 ∀𝑘 > 0. We need to calculate how

many steps it take so that 𝑦𝑘 < 0. Again from (2.3.3) 𝑦𝑘+1 − 𝑦𝑘 is decreasing, which means that 𝑦𝑘

is decreasing faster and faster. So it takes at most

𝑦0
𝑦0 − 𝑦1

=

1
𝑞+1
𝑛−(𝑛−2)𝑞
(𝑞+1)(𝑛−1)

=

𝑛 − 1
𝑛 − (𝑛 − 2)𝑞

steps to make 𝑥𝑘 negative. Let 𝐾 (𝑛, 𝑞) be the least integer larger than

what we want in the lemma.

Use (2.3.2) and do induction, we have

(2.3.4)

𝑛−1

𝑛−(𝑛−2)𝑞 , and this 𝐾 (𝑛, 𝑞) is
□

∥𝑢∥𝑥𝑘+2 ≤ 𝐶 (∥𝑢∥𝑥𝑘+1 + ∥𝑢∥
(cid:16)

≤ 𝐶

(∥𝑢∥𝑥𝑘 + ∥𝑢∥

𝑞
𝑥𝑘+1)

𝑞
𝑥𝑘 ) + (∥𝑢∥𝑥𝑘 + ∥𝑢∥

𝑞

𝑥𝑘 )𝑞(cid:17)

(cid:16)

≤ 𝐶

(∥𝑢∥𝑥𝑘 + ∥𝑢∥

𝑞
𝑥𝑘 ) + 2𝑞 (∥𝑢∥

𝑞
𝑥𝑘 + ∥𝑢∥

(cid:17)

𝑞2
𝑥𝑘 )

(2.3.5)

≤ 𝐶 (∥𝑢∥𝑥𝑘 + ∥𝑢∥

𝑞2
𝑥𝑘 )

· · ·

≤ 𝐶 (∥𝑢∥𝑥0 + ∥𝑢∥

𝑞𝑘+2
𝑥0

)

where in the third line we used the inequality (𝑎 + 𝑏)𝑞 ≤ 2𝑞 (𝑎𝑞 + 𝑏𝑞).

∥𝑢∥∞ ≤ 𝐶 (𝑀, 𝑛, 𝑞) ∥𝑢∥𝑥0

(2.3.6)

41

Remark 2.3.1. Note that if we 𝑞 is bounded away from 𝑛

𝑛−2, then from (2.3.4) we can have a uniform
bound for 𝐾 that doesn’t depend on 𝑞. However, the constant 𝐶 in (2.3.2) does depends on 𝑞 and

we fails to get a universal estimate. If we could find a universal constant, then (2.3.6) becomes

∥𝑢∥∞ ≤ 𝐶𝜆𝑁 ∥𝑢∥𝑥0

where both 𝑁 and 𝐶 are independent of 𝑞. And (2.2.13) is

∫

𝜇

𝑀

|∇𝑢|2 ≤ 𝜆

∫

𝑀

(𝑞𝑢𝑞−1 − 1)|∇𝑢|2

≤ 𝐶𝜆(𝑞(𝐶𝜆) 𝑁 (𝑞−1) − 1)

∫

𝑀

|∇𝑢|2

We will be able to track how the critical 𝜆𝑞 changes with 𝑞, and it can easily seen that 𝜆𝑞 → ∞ as

𝑞 → 1.

Remark 2.3.2. One might inquire whether the bounds established in these two sections can be

made universal when Ricci curvature and the second fundamental form are bounded below. Our

interest lies in understanding how geometric conditions impact the solutions of PDEs. However,

unlike the Laplacian operator, obtaining estimates for the Dirichlet-to-Neumann operator (𝐷𝑁)

proves challenging.

For instance, a comparison theorem for the heat kernel in terms of Ricci curvature is established

in [CY81] and [LY86]. Under Ricci curvature restrictions, both lower and upper bounds for

eigenvalues of the Laplacian operator can be derived. For further details, see Chapter 3 of [SY94]

or [Li12]. However, these methods cannot be directly extended to the Dirichlet-to-Neumann

operators. The lack of knowledge regarding how geometric conditions affect Dirichlet-to-Neumann

operators poses a significant challenge in Conjecture 2.1.

2.4 Estimate of the Infimum

In the previous two sections we derived 𝐿∞ estimate and then non-existence theorem for

(2.2.8). Note that (2.2.8) comes from the Euler-Lagrangian equation of functional (2.1.2, so the

non-existence theorem gives us 𝑠(𝑞, 𝜆) = 𝜆|Σ|

1−𝑞
1+𝑞 for certain (𝑞, 𝜆)’s. If we closely examine the

42

functional, we could get better estimate.

Let (𝑀, Σ, 𝑔) be an arbitrary manifold with boundary. Throughout this section ∇ and ∇ denote

gradient on 𝑀 and Σ respectively. Integration without lower indices denotes integration on Σ. We

begin with a lemma:

Lemma 2.4.1. For 1 < 𝑞 ≤ 𝑞0 < 𝑛+1

𝑛−1, and ∫ 𝑓 𝑞+1 = |Σ|, there exist a constant 𝐶 that only depends

on the metric and 𝑞0 so that

∫

Σ

𝑓 2𝑞 (log 𝑓 )2 ≤ 𝐶

∫

Σ

|∇ 𝑓 |2

(2.4.1)

Proof. We first show that 𝑥2𝑞 (log 𝑥)2 ≤ 𝐶 ((𝑥 −1)2 + |𝑥 −1|2𝑞1) where 𝑞1 is chosen to lie in (𝑞0, 𝑛+1
𝑛−1)

and 𝐶 only depends on 𝑞0. This could be seen by looking into the following three cases.

𝑖)𝑥 ∈ (0,

𝑖𝑖)𝑥 ∈ [

1
2

) : 𝑥2𝑞 (log 𝑥)2 is bounded above and (𝑥 − 1)2 is bounded below;

1
2
, 2] : 𝑥2𝑞 is bounded above and (log 𝑥)2 is bounded by (𝑥 − 1)

𝑖𝑖𝑖)𝑥 ∈ (2, ∞) : 𝑥2𝑞 (log 𝑥)2 ≤ 𝑥2𝑞0 (log 𝑥)2 and therefore uniformly bounded by |𝑥 − 1|2𝑞1.

Then we need to bound ∫ ( 𝑓 − 1)2 and ∫ | 𝑓 − 1|2𝑞1 by ∫
Σ
to “modify" the power for ∫ | 𝑓 − 1|2𝑞1. Namely, use 𝜃 = 1
𝑞1

|∇ 𝑓 |2. We might apply Hölder inequality

in Hölder inequality:

∫

| 𝑓 − 1|2𝑞1 ≤ ∥ 𝑓 − 1∥

2𝑞1𝜃
𝑞2

∥ 𝑓 − 1∥

2(1−𝜃)𝑞1
2

≤ 𝐶 ∥ 𝑓 − 1∥2
𝑞2

(2.4.2)

𝑛−3 . This is where we use the assumption that 𝑞0 < 𝑛+1
where 𝑞2 = 2
𝑛−1. So 𝑞2 is strictly
2−𝑞1
below the Sobolev conjugate. The last inequality follows from ∫ 𝑓 𝑞+1 = |Σ| and Hölder inequality

< 2(𝑛−1)

again, and it’s easy to see that 𝐶 can be chosen independent of 𝑞 < 𝑞1. Now it suffices to show that
∥ 𝑓 − 1∥𝑞2 ≤ 𝐶 ∥∇ 𝑓 ∥2 for 𝑓 satisfying ∫ 𝑓 𝑞+1 = 𝐴(Σ). This comes from a generalized Poincarè
∫ 𝑢 defined as the average of 𝑢 over Σ, then
inequality. Let

⨏

𝑢 := 1
|Σ|

⨏

∥ 𝑓 − (

𝑓 𝑟)

1

𝑟 ∥𝑞2 ≤ 𝐶 ∥∇ 𝑓 ∥2

(2.4.3)

for 𝑟 < 𝑞0 + 1 < 2(𝑛−1)
𝑛−3

and 𝐶 only depends on 𝑞1 and 𝑞2.(In our case and 𝑞2 depends on

𝑞0, so 𝐶 only depends on 𝑞0). The proof is by contradiction and modified from the standard

43

⨏

proof. Suppose (2.4.3) is not true, and we can find a sequence 𝑟𝑖 and 𝑓𝑖 so that ∥∇ 𝑓𝑖 ∥2 → 0 and

∥ 𝑓𝑖 − (

converges in 𝐿𝑞2. Since ∥∇ 𝑓𝑖 ∥2 → 0, the sequence 𝑓𝑖 − (

𝑓 𝑟𝑖
𝑖 )1/𝑟𝑖 ∥𝑞2 = 1. By compactness, we might pick a subsequence so that 𝑓𝑖 − (

𝑓 𝑟𝑖
𝑖 )1/𝑟𝑖
𝑓 𝑟𝑖
𝑖 )1/𝑟𝑖 converges in 𝐻1 to a constant
function. So we might write 𝑓𝑖 = 𝑎𝑖 + ℎ𝑖 where 𝑎𝑖 are constants and ℎ𝑖 → 0 in 𝐻1. As a
ℎ𝑟𝑖
𝑖 )1/𝑟𝑖 by triangle inequality and it follows that
𝑖 )1/𝑟𝑖 ∥𝑞2 = 1. ∫ | 𝑓 − 1|2
𝑓 𝑟𝑖
□

𝑓 𝑟𝑖
𝑖 )1/𝑟𝑖 ≤ 𝑎𝑖 + (
ℎ𝑟𝑖
𝑖 )1/𝑟𝑖 . This contradicts with ∥ 𝑓𝑖 − (

ℎ𝑟𝑖
𝑖 )1/𝑟𝑖 ≤ (
𝑓 𝑟𝑖
𝑖 )1/𝑟𝑖 | ≤ |ℎ𝑖 | + (

can be estimated in a similar way.

result, 𝑎𝑖 − (
⨏

| 𝑓𝑖 − (

⨏

⨏

⨏

⨏

⨏

⨏

⨏

In the lemma above, the base point is 1 and we measured distance from 𝑓 to 1. That’s why

we have the three cases in the proof above.

In order to apply this lemma we need a different

normalization from (2.2.8).

on 𝑀 𝑛

= −𝜆𝑢 + 𝑟 (𝑞, 𝜆)𝑢𝑞 on Σ𝑛−1

(2.4.4)

Δ𝑢 = 0
𝜕𝑢
𝜕𝜈
𝑢𝑞+1 = |Σ|

∫

Σ

If we assume 𝑢 is a minimzer for 𝑠(𝑞, 𝜆), multiply this equation by 𝑢 and integrate by parts, it’s

easy to see that 𝑠(𝑞, 𝜆) = 𝑟 (𝑞, 𝜆)|Σ|

𝑞−1
𝑞+1 and thus

𝜆 ≥ 𝑟 (𝑞, 𝜆)

(2.4.5)

Theorem 2.7. For 1 < 𝑞 ≤ 𝑞0 < 𝑛+1

𝑛−1, there exists a constant 𝐶 depending only on 𝑛, 𝑞0 and the

metric so that 𝑠(𝑞, 𝜆) = 𝜆|Σ|

1−𝑞
1+𝑞 provided

𝜆(𝑞 − 1) < 𝐶

(2.4.6)

This method only works for 𝑞 < 𝑛+1

𝑛−1. The case 𝑞 < 𝑛

𝑛−2 will be dealt in the next section from a

different viewpoint.

We’ll need the following lemma:

44

Lemma 2.4.2 (Pohozaev ideneity). Let (𝑀, Σ, 𝑔) be a compact Riemannian manifold with bound-

ary, and 𝑔′ = 𝑔|Σ. Suppose 𝑢 is a smooth function and 𝑋 is a smooth vector field. Then

∫

𝑀

⟨∇∇𝑢 𝑋, ∇𝑢⟩ −

1
2

|∇𝑢|2div𝑔 𝑋 + (𝑋𝑢)Δ𝑢 =

∫

Σ

𝜕𝑢
𝜕𝑛

(

⟨𝑋, ∇𝑢⟩ −

1
2

|∇𝑢|2⟨𝑋, (cid:174)𝑛⟩)

(2.4.7)

Proof. By direct calculation

div(𝑋𝑢)∇𝑢 = ⟨∇∇𝑢 𝑋, ∇𝑢⟩ + ⟨𝑋, ∇∇𝑢∇𝑢⟩ + (𝑋𝑢)Δ𝑢

div(

1
2

|∇𝑢|2𝑋) =

1
2

|∇𝑢|2div𝑋 + ⟨∇𝑢, ∇𝑋 ∇𝑢⟩ =

1
2

|∇𝑢|2div𝑋 + ⟨𝑋, ∇∇𝑢∇𝑢⟩

By taking the difference and applying integration by parts to the left-hand side, we obtain the

desired equality.

□

Proof. Throughout this proof 𝐶 denotes some constants that only depends on 𝑞0 and the metric 𝑔.

And it might change from line to line. By Pohozaev identity (2.4.7) for harmonic function 𝑢 and

arbitrary smooth vector field 𝑋 we have

∫

𝑀

1
2

(⟨ ¯∇ ¯∇𝑢 𝑋, ¯∇𝑢⟩ −

| ¯∇𝑢|2div𝑔 𝑋) =

𝜕𝑢
𝜕𝑛
(cid:12)Σ = (cid:174)𝑛 in this equality and note that ¯∇𝑋 is bounded by compactness,

| ¯∇𝑢|2⟨𝑋, (cid:174)𝑛⟩)

⟨𝑋, ∇𝑢⟩ −

1
2

Σ

(

∫

Fix a 𝑋 satisfying 𝑋(cid:12)

𝜕𝑢
𝜕𝑛
𝜕𝑢
𝜕𝑛

∫

)2 +

(| ¯∇𝑢|2div𝑔 𝑋 − 2⟨ ¯∇ ¯∇𝑢 𝑋, ¯∇𝑢⟩)
𝑀
∫
)2 + 𝐶

| ¯∇𝑢|2

𝑀

(2.4.8)

∫

Σ

|∇𝑢|2 =

≤

∫

Σ
∫

(

(

Σ
∫

≤ 𝐶

𝜕𝑢
𝜕𝑛

(

)2

Σ

The last inequality follows by lemma 2.2.4. Adding 𝜆2−𝑟 2

𝜆

∫
𝑀 | ¯∇𝑢2| ≥ 0 (by (2.4.5)) to the right

hand side and using (2.4.4), it becomes the following

∫

|∇ 𝑓 |2 ≤ 𝐶 (

∫

𝜕𝑢
𝜕𝑛

)2 +

𝜆2 − 𝑟 2
𝜆

∫

∫

𝑓 2𝑞 − 2𝜆𝑟

∫

(𝑟

𝑓 𝑞+1 − 𝜆

𝑀

| ¯∇𝑢2|)
𝑓 𝑞+1 + 𝜆2∫
∫

𝑓 2)]

(
= 𝐶 [(𝑟 2∫
𝜆2 − 𝑟 2
𝜆
∫

+

= 𝐶 [𝑟 2(

𝑓 2𝑞 + 𝑓 2) + 𝑟 (

𝜆2 − 𝑟 2
𝜆

∫

− 2𝜆)

𝑓 𝑞+1]

45

𝑓 2)

(2.4.9)

Consider the function 𝜙(𝑥) = 𝑎𝑞+1+𝑥 + 𝑎𝑞+1−𝑥 for 𝑥 ∈ [0, 𝑞 − 1] and 𝑎 > 0. Taylor expansion implies

that for some 𝜃 ∈ [0, 𝑞 − 1]

𝜙(𝑞 − 1) = 2𝑎𝑞+1 +

𝜙′′(𝜃)
2

(𝑞 − 1)2 =2𝑎𝑞+1 +

≤2𝑎𝑞+1 +

(𝑞 − 1)2
2
(𝑞 − 1)2
2

(log 𝑎)2𝑎𝑞+1(𝑎𝜃 + 𝑎−𝜃)

(log 𝑎)2(𝑎2𝑞 + 𝑎2)

Using this estimate in 2.4.9,

∫

|∇ 𝑓 |2 ≤ 𝐶 [

(𝑞−1)2
2

∫

𝑟 2(

( 𝑓 2𝑞 + 𝑓 2) log 𝑓 +𝑟 (

−𝑟 2
𝜆

∫

+2𝑟 −𝜆)

𝑓 𝑞+1]

∫

𝐶
2

(𝑞 − 1)2𝑟 2(
≤
≤ 𝐶 (𝑞 − 1)2𝑟 2 ∫

𝑓 2𝑞 (log 𝑓 )2 +

∫

𝑓 2(log 𝑓 )2)

(2.4.10)

𝑓 2𝑞 (log 𝑓 )2

Using Lemma 2.4.1 and (2.4.5), (2.4.10) becomes

∫

|∇ 𝑓 |2 ≤ 𝐶 (𝑞 − 1)2𝜆2 ∫

|∇ 𝑓 |2

and 𝑓 , and therefore 𝑢, will be constant if (𝑞 − 1)𝜆 is small.

□

Remark 2.4.1. Note that (2.4.8) was obtained in (2.2.3) by directly utilizing the ellipticity of the

Dirichlet-to-Neumann operator. However, it is challenging to discern how curvature conditions

come into play in that method. (2.4.8) is more likely to be connected to geometry, and the problem

is how to construct a nice vector field. This provides some information, but not precisely what we

are seeking. This idea will come back later in a log-Sobolev inequality.

One might hope to get a uniform bound for the constant in Dirichlet-to-Neumann operator, but

such an estimate doesn’t exist even under the condition of positive sectional curvature and 𝐼 𝐼 ≥ 1

where 𝐼 𝐼 is the second fundamental form. Actually, consider the ellipse

{(𝑥, 𝑦)(cid:12)

(cid:12)𝑥2 + 𝑘 2𝑦2 ≤ 1}

Under scaling of the metric ¯𝑔 = 𝑘 −2𝑔, the second fundamental form can be arbitrarily large. At the
same time both | ¯∇ 𝑓 |2 = 𝑘 2|∇ 𝑓 |2 and ( 𝜕𝑢
𝜕𝑛 )2 are scaled by the same factor. So we might

𝜕 ¯𝑛 )2 = 𝑘 2( 𝜕𝑢

46

forget about the restriction on the 𝐼 𝐼. The normal vector and tangent vector are (cid:174)𝑛 = (𝑥, 𝑘 2𝑦) and

(cid:174)𝑣 = (𝑘 2𝑦, 𝑥) respectively. For 𝑢 = 𝑥 we calculate as follows

∫

𝜕𝑢
𝜕𝑛

(

)2 = 4

((1, 0) ·

√︄

(cid:174)𝑛
| (cid:174)𝑛|

)2

𝑘 4𝑦2 + 𝑥2
𝑘 4𝑦2

dx

∫ 1

0
∫ 1

= 4

𝑥2
𝑘√︁(𝑘 2(1 − 𝑥2) + 𝑥2) (1 − 𝑥2)
4
𝑘
𝜕𝑛 )2 = ∫ 1 ≥ 4. So ∫ |∇𝑢|2 can’t be uniformly bounded by ∫ ( 𝜕𝑢

0
∫ 1

1 − 𝑥2

4
𝑘

dx

√

≤

=

𝑥

0

(2.4.11)

Also note that ∫ |∇𝑢|2 + ( 𝜕𝑢
curvature and 𝐼 𝐼 restriction. Consider 𝑢 = 𝑦, we see that ∫ ( 𝜕𝑢
∫ |∇𝑢|2, either.

𝜕𝑛 )2 under
𝜕𝑛 )2 can’t be uniformly bounded by

Remark 2.4.2. The idea of this proof comes from the paper by Ou and Lin [LO23]. We translate
|∇ 𝑓 |2 can be bounded by combination of ∫
their work as follows: ∫
𝑀 | ¯∇𝑢|2. These two
terms are “difference" of 𝐿 𝑝 norms by (2.2.8), and this “difference" can be bounded by ∫
|∇ 𝑓 |2.
In [LO23], this “difference" is measured by the ratio of 𝐿 𝑝 norm. We treated it differently by taking

𝜕𝑛 )2 and ∫
( 𝜕𝑢

Σ

Σ

Σ

the subtraction.

2.5 An ODE Approach

Consider 𝑠𝜆,𝑞 − 𝐴(Σ)

𝑞−1
𝑞+1 𝜆 ≤ 0.

If this inequality is strict, 𝐸𝑞,𝜆 must have a non-constant

minimizer. Using this minimizer, we are going to show that the strict negativity is preserved along

some curve of 𝜆, 𝑞 which looks like (2.5.1) in the theorem below. But we know from the work

of [GW20] that for unit ball, 𝑞 = 𝑛

𝑛−2, 𝜆 = 𝑛−2

2 , 𝐸𝑞,𝜆 only admits constant minimizer, which gives

some restriction on 𝑞, 𝜆.

Lemma 2.5.1. Let (𝑀, Σ, 𝑔) be a Riemannian manifold with boundary so that 𝐴(Σ) = 1. Suppose

𝑠(𝑞0, 𝜆0) − 𝜆0 ≤ −𝜖 < 0 for some (𝜆0, 𝑞0), then this inequality remains valid along the curve

𝑞 + 1
𝑞 − 1

= 𝐶 (𝜆 − 𝜖)

47

(2.5.1)

for 𝜆 ≤ 𝜆0, where 𝐶 =

𝑞0+1

(𝑞0−1)(𝜆0−𝜖) is chosen so that (𝜆0, 𝑞0) is on the curve.

Proof. Along the curve (2.5.1), 𝑠(𝜆) and 𝑞(𝜆) are functions of 𝜆 only.

It suffices to show that

for 𝜆1 satisfying 𝑠(𝜆1) = 𝜆1 − 𝜖, 𝑠(𝜆) − 𝜆 will be decreasing along the curve in the −𝜆 direction

near 𝜆1. Note that 𝑠(𝜆1) = 𝜆1 − 𝜖 implies the existence of a non-constant minimizer for 𝐸𝑞1,𝜆1

satisfying (2.2.8), where 𝑞1 = 𝑞(𝜆1). Denote it by 𝑢. Fix this 𝑢, and we want to show that

𝐸𝑞(𝜆),𝜆 (𝑢) decreases fast enough in −𝜆 direction along the curve (2.5.1), and the theorem follows

since 𝑠(𝑞(𝜆, 𝜆) ≤ 𝐸𝜆,𝑞(𝜆) (𝑢) for all 𝜆. Namely, we need to prove the following inequality:

𝜕
𝜕 (−𝜆)

(𝐸𝜆,𝑞(𝜆) (𝑢) − 𝜆) < 0

One calculates that

𝜕
𝜕𝑞

(

∫

Σ

We assumed that

𝑓 𝑞+1)

2
𝑞+1 = (

∫

𝑓 𝑞+1)

2
𝑞+1 [−

Σ
𝑓 𝑞1+1 = 𝐴(Σ) = 1, so

⨏

Σ

2
(𝑞 + 1)2

∫

log

Σ

𝑓 𝑞+1 +

∫
Σ

2
𝑞 + 1

𝑓 𝑞+1 log 𝑓
∫
Σ

𝑓 𝑞+1

]

𝜕
𝜕𝜆

𝐸𝜆,𝑞(𝜆) (𝑢)(cid:12)
(cid:12)𝜆1

=

∫

Σ

𝑓 2 −

2𝑠(𝜆1)
𝑞1 + 1

𝑞′(𝜆1)

∫

Σ

𝑓 𝑞1+1 log 𝑓

So it suffices to show

By Hölder inequality and 𝐴(Σ) =
𝑓 𝑞+1 log 𝑓 ≥ 0. Also note that ∫
∫
Σ
Σ

∫

Σ

𝑓 2 −
⨏

∫

𝑞′(𝜆1)

𝑓 𝑞+1 log 𝑓 > 1

2𝑠(𝜆1)
𝑞 + 1
𝑓 𝑞+1 = 1, we have ∫
Σ
Σ
𝑓 𝑥 is a strict convex function in 𝑥, we have

Σ

𝑓 𝑞+1+𝜖 ≥ 1 for 𝜖 > 0 and it follows that

(2.5.2)

∫

Σ

𝑓 2 + (𝑞 − 1)

∫

Σ

𝑓 𝑞+1 log 𝑓 >

∫

Σ

𝑓 𝑞+1 = 1

This inequality is strict since 𝑢 is not constant. So (2.5.2) holds provided

−2𝑞′(𝜆) (𝜆 − 𝜖)
𝑞 + 1

≥ 𝑞 − 1

48

The solution for the equality is exactly (2.5.1), and we finish the proof.

□

Corollary 2.1. For (B𝑛, S𝑛−1, 𝑔) the standard metric, 𝑞 ≤ 𝑛

𝑛−2, 𝑠(𝑞, 𝜆) is achieved only by constant

functions for

(𝑞 − 1) (𝜆 −

𝑛 − 2
2(𝑛 − 1)

) <

𝑛 − 2
𝑛 − 1

(2.5.3)

Proof. We will first scale the metric so that the area of the boundary is 1, namely consider

(B𝑛, S𝑛−1, 𝑔 = 𝑘 2𝑔) where 𝑘 = 𝐴(S𝑛−1)

1

1−𝑛 .The function 𝐸𝑞,𝜆 (𝑢) changes as follows

𝐸 𝑞,𝜆 (𝑢) =

∫
B𝑛 |∇𝑔𝑢|2dVol𝑔 + 𝜆 ∫
S𝑛−1 𝑢𝑞+1dS𝑔)

(∫

2
𝑞+1

S𝑛−1 𝑢2dS𝑔

= 𝑘 𝑛−2−

2(𝑛−1)
𝑞+1

B𝑛 |∇𝑔𝑢|2dVol𝑔 + 𝑘𝜆 ∫
∫
S𝑛−1 𝑢𝑞+1dS𝑔)

(∫

2
𝑞+1

S𝑛−1 𝑢2dS𝑔

= 𝑘 𝑛−2−

2(𝑛−1)
𝑞+1 𝑄 𝑘𝜆,𝑞 (𝑢)

(2.5.4)

By the work of ([GW20]) for unit ball, (2.2.8) admit only constant solutions for 𝜆 = 𝑛−2

After the scaling (2.5.4), for ¯𝑔 = 𝑘 2𝑔 this critical point becomes (𝑞, 𝜆) = ( 𝑛

2 , 𝑞 = 𝑛
𝑛−2.
2𝑘 ). Suppose

𝑛−2, 𝑛−2

¯𝑠𝑞0,𝜆0 < 𝜆0 for some (𝜆0, 𝑞0) satisfying

𝜆1 <

(𝑛 − 2) (𝑞 + 1)
2𝑘 (𝑛 − 1) (𝑞 − 1)

(2.5.5)

When 𝜖 is small enough, ¯𝑠𝑞0,𝜆0 < 𝜆0 − 𝜖. And by Theorem 2.5.1 this inequality remains valid
along (2.5.1) for 𝜆 < 𝜆0. In particular, we let 𝑞 = 𝑛

𝑛−2, then

𝜆 =

𝑞 + 1
𝐶 (𝑞 − 1)

+ 𝜖 = (𝑛 − 1)

(𝑞0 − 1) (𝜆0 − 𝜖)
𝑞0 + 1

+ 𝜖

(2.5.6)

If (2.5.5) holds, we can make 𝜖 small so that 𝜆 < 𝑛−2

2𝑘 , which is contradiction since we must have

¯𝑠 𝑛−2

2𝑘 , 𝑛
𝑛−2

= 𝑛−2

2𝑘 . Now transfer this result back to the unit ball and the proof is finished.

□

49

Remark 2.5.1. For standard balls Wang’s conjecture been completely solved in [GL23], and the

theorem above is also included.

In the preceding sections, we demonstrated that for 𝑞 < 𝑛

𝑛−2, there exists a corresponding 𝜆 such
that 𝑠(𝑞, 𝜆) is only achieved by constants. Let’s fix one such pair as (𝑞0, 𝜆0) and employ a similar

argument to the one in the corollary above. This will yield a result similar to Theorem 2.7, but with

𝑞0 < 𝑛

𝑛−2 instead of 𝑞0 < 𝑛+1

𝑛−1. These two approaches are distinct and offer different perspectives

on Wang’s conjecture.

Remark 2.5.2. The proof of the corollary does not rely on the specific structure of the manifold.

The only instance where (B𝑛, S𝑛−1, 𝑔) is involved is at the critical point ( 𝑛

𝑛−2, 𝑛−2

2 ). Therefore,

our method is applicable to any manifold as long as one can compute such a critical point. The

challenge lies in determining how to obtain such a point under curvature restrictions. When

𝑞 = 𝑛

𝑛−2, the problem is related to type II Yamabe problem. A breakthrough in Yamabe problem

might help us find a critical (𝑞, 𝜆).

2.6 Critial Power Case

If 𝑞 = 𝑛

𝑛−2, the trace operator is only continuous and fails to be compact, making the existence of
minimizers more challenging. In this section, I will derive some existence results for the minimizer

of 𝑠 (cid:0) 𝑛

𝑛−2, 𝜆(cid:1). It’s difficult to determine whether these minimizers become constants for small 𝜆’s.

Lemma 2.6.1. For any compact Riemannian manifold with boundary and 𝜆 ≥ 0,

4(𝑛 − 1)
𝑛 − 2

𝑠(

𝑛
𝑛 − 2

, 𝜆) ≤ 𝑌 (B𝑛, S𝑛−1, 𝑑𝑥2)

(2.6.1)

Proof. Fix a point 𝑝 ∈ Σ. We can find a small neighborhood 𝑈 of 𝑝 ∈ 𝑀 so that 𝑈 = 𝐵𝑛−1(𝛿)×(0, 𝛿)

for some small 𝛿. Fix a cut-off function 𝜙 so that 𝜙 = 1 in 𝐵𝑛−1(𝛿/2) × (0, 𝛿/2) and vanishes

outside 𝑈. Let {𝑥𝑖}𝑛−1

𝑖=1 be the coordinates for 𝐵𝑛−1(𝛿) and 𝑡 coordinate for (0, 𝛿). Define

𝑣𝜖 = (

𝜖
(𝜖 + 𝑡)2 + |𝑦|2 )

𝑛−2
2

(2.6.2)

50

Recall that we can use 𝜙𝑣𝜖 as test functions to establish the type II Yamabe inequality (1.2.17).

𝑅𝑢2𝑑𝑉 and ∫
Σ

∫
𝑀
we can demonstrate that ∫
𝑀

(cid:16) 𝑛−2
2(𝑛−1)

𝐻 − 𝜆

The differences between the functional of the type II Yamabe problem and 𝐸 𝑛

𝑛−2 ,𝑞 lie in the terms
𝑢2𝑑𝑆. Note that 𝐻 and 𝑅 remain bounded for a fixed metric. If

(cid:17)

𝑢2𝑑𝑉 and ∫
Σ
complete. Let 𝑑𝑉𝐸 and 𝑑𝑆𝐸 be the volume form with respect to Euclidean space.

𝑢2𝑑𝑉 vanish as 𝜖 → 0 for 𝑢 = 𝜙𝑣𝜖 , then the proof is

∫

𝑀

(𝜙𝑣𝜖 )2𝑑𝑉𝑔 ≤

∫

𝑣2
𝜖 𝑑𝑉𝑔

𝑈
∫

≤ 𝐶

𝑣2
𝜖 𝑑𝑉𝐸

𝑈
∫ 𝛿

0
∫ 𝛿

0
∫ 𝛿

0
∫ 𝛿

𝜖

0

= 𝐶

= 𝐶

≤ 𝐶

= 𝐶

∫

𝐵𝑛−1 (𝛿)

∫

𝑡+𝜖 )

𝐵𝑛−1 ( 𝛿
𝜖 𝑛−2
(𝑡 + 𝜖)𝑛−3
𝜖 2
(1 + 𝑡)𝑛−3

𝑑𝑡

𝑑𝑡 ≤ 𝐶𝜖 2

(

𝜖

(𝜖 + 𝑡)2 + |𝑦|2 )𝑛−2𝑑𝑦𝑑𝑡

𝜖 𝑛−2
(𝑡 + 𝜖)𝑛−3

1
(1 + |𝑧|2)𝑛−2

𝑑𝑧𝑑𝑡

We used change of variable in the third and fifth line. Similarly, we have

∫

Σ

(𝜙𝑣𝜖 )2𝑑𝑆𝑔 ≤

∫

𝑣2
𝜖

Σ
∫

≤ 𝐶

𝐵𝑛−1 (𝛿)

𝑣2
𝜖 𝑑𝑆𝐸

(

𝜖

𝜖 2 + |𝑦|2 )𝑛−2𝑑𝑧

∫

𝐵𝑛−1

∫

= 𝐶

= 𝐶

𝐵𝑛−1 ( 𝛿
𝜖 )

𝜖
(1 + |𝑧|2) (𝑛−2)

𝑑𝑧 ≤ 𝐶𝜖

(𝜙𝑣𝜖 )

2(𝑛−1)
𝑛−2 ≥

∫

Σ

∫

2(𝑛−1)
𝑛−2
𝜖

𝑣

𝑑𝑆𝑔

𝐵𝑛−1 (𝛿/2)
∫

𝐵𝑛−1 (𝛿/2)

∫

≥ 𝐶

= 𝐶

𝜖

(

𝜖 2 + |𝑦|2 ) (𝑛−1) 𝑑𝑆𝐸
1 + |𝑧|2 ) (𝑛−1) 𝑑𝑧 ≥ 𝐶

1

(

𝐵𝑛−1 (𝛿/(2𝜖))
Let 𝜖 → 0, and the three estimates prove the lemma.

If 𝑞 < 𝑛

𝑛−2, such a bound doesn’t exist.

51

(2.6.3)

□

Lemma 2.6.2. 𝑠(𝑞, 𝜆) → ∞ as 𝜆 → ∞ for 𝑞 < 𝑛

𝑛−2.

Proof. We prove by contradiction. Suppose not. Then there exists 𝜆𝑖 → and 𝑢𝑖 such that 𝐸𝑞,𝜆𝑖 (𝑢𝑖) <
𝐶. Without loss of generality, we might assume ∥𝑢𝑖 ∥ 𝐿𝑞+1 (Σ) = 1. Then we must have ∫
𝑀 |∇𝑢𝑖 |2 < 𝐶
and ∫
𝑢2
𝑖 → 0. By Alauglu theorem we can find a subsequence, still denoted by 𝑢𝑖, such that
Σ
𝑛−2, by compactness, we can pick a further
□

subsequence so that ∥𝑢0∥ 𝐿𝑞+1 (Σ) = lim ∥𝑢𝑖 ∥ 𝐿𝑞+1 (Σ) = 1, which is a contradiction.

𝑖 = 0. Since 𝑞 < 𝑛
𝑢2

𝑢𝑖 ⇀ 𝑢0 and ∫
Σ

0 = lim ∫
𝑢2

Σ

According to the computations in Lemma 2.6.1, we have 𝑣𝜖 → 0 in 𝐿2(Σ), but they do not

2(𝑛−1)

converge in 𝐿

where 𝑠( 𝑛

𝑛−2 (Σ). This elucidates why the argument fails for 𝑞 = 𝑛

𝑛−2. In this critical case,
𝑛−2, 𝜆) is bounded in 𝜆, the dynamics are quite different. The key observation is that if
𝑛−2, 𝜆) stops increasing for large 𝜆, it is likely minimized through a sequence of functions that
blow up somewhere, with their 𝐿2(Σ) norms tending to zero. This phenomenon is akin to what

𝑠( 𝑛

is observed in (2.6.2). Consequently, 𝑠( 𝑛

𝑛−2, 𝜆) admits no minimizer in this scenario, not even
𝑛−2, 𝜆) keeps increasing in 𝜆, functions like (2.6.2) are ruled out
as minimizing sequences. This exclusion opens up the possibility of obtaining a minimizer. These

constants. On the contrary, if 𝑠( 𝑛

observations can be made concrete by the following theorem.

Theorem 2.8. i):If there exists 𝜇 < 𝜆 such that 𝑠( 𝑛

𝑛−2, 𝜆) = 𝑠( 𝑛

𝑛−2, 𝜇), then 𝑠( 𝑛

𝑛−2, 𝜆) doestn’t admit

any minimizer.

ii): If there exists 𝜆 < 𝜇 such that 𝑠( 𝑛

𝑛−2, 𝜆) < 𝑠( 𝑛

𝑛−2, 𝜇), then 𝑠( 𝑛

𝑛−2, 𝜆) admits a minimizer.

Proof. Part i):

Suppose 𝑠( 𝑛

𝑛−2, 𝜆) admit a minimizer 𝑢. Then

𝑠(

𝑛
𝑛 − 2

, 𝜇) ≤ 𝐸 𝑛

𝑛−2 ,𝜇 (𝑢) < 𝐸 𝑛

𝑛−2 ,𝜆 (𝑢) = 𝑠(

𝑛
𝑛 − 2

, 𝜆)

which contradicts our assumption.

Part ii):

Let 𝑢𝑖 be a minimizing sequence for 𝑠( 𝑛

𝑛−2, 𝜆). We can scale 𝑢𝑖 so that ∥𝑢𝑖 ∥ 𝐿 𝑝 = 1, where 𝑝 =

2(𝑛−1)
𝑛−2 .

52

Then

lim
𝑖→∞

(cid:16) ∫

𝑀

|∇𝑢𝑖 |2 + 𝜆

∫

Σ

(cid:17)

𝑢2
𝑖

= 𝑠(

𝑛
𝑛 − 2

, 𝜆)

Use these 𝑢𝑖 as test-function for 𝜇, and we have

∫

𝑀

|∇𝑢𝑖 |2 + 𝜇

∫

Σ

𝑢2
𝑖 ≤ 𝑠(

𝑛
𝑛 − 2

, 𝜇)

By Alaoglu theorem and compactness we can get a subsequence so that

𝑢𝑖 ⇀ 𝑢 in 𝐻1(𝑀), 𝑢𝑖 ⇀ 𝑢 in 𝐿 𝑝 (Σ),

𝑢𝑖 → 𝑢 in 𝐿2, 𝑢𝑖 → 𝑢 a.e. in 𝑀

Use these 𝑢𝑖 as test-function for 𝜇,

∫

𝑀

|∇𝑢𝑖 |2 + 𝜇

∫

Σ

𝑢2
𝑖 ≥ 𝑠(

𝑛
𝑛 − 2

, 𝜇)

Take the difference between the (2.6.4) and (2.6.5), and we get

(2.6.4)

(2.6.5)

(2.6.6)

(𝜇 − 𝜆)

∫

Σ

𝑢2 = lim(𝜇 − 𝜆)

∫

Σ

𝑢2
𝑖 ≥ 𝑠(

𝑛
𝑛 − 2

, 𝜇) − 𝑠(

𝑛
𝑛 − 2

, 𝜆) > 0

This is where we used our assumption. This rules out possibility that 𝑢 ≡ 0, which happens in the

proof of lemma 2.6.1.

Next we are going to show 𝑢 minimizes 𝑠( 𝑛

𝑛−2, 𝜆). Since 𝑢𝑖 ⇀ 𝑢 in 𝐿 𝑝 (Σ), we have ∥𝑢∥ 𝑝 ≤ 1. Let

𝑣𝑖 = 𝑢𝑖 − 𝑢. By a result in [BL83],

1 = lim ∥𝑢𝑖 ∥

𝑝
𝑝 = lim ∥𝑢 + 𝑣𝑖 ∥

𝑝
𝑝 = ∥𝑢∥

𝑝
𝑝 + lim ∥𝑣𝑖 ∥

𝑝
𝑝

Note that 𝑢 ≠ 0, so ∥𝑢∥ 𝑝 ≤ 1, lim ∥𝑣𝑖 ∥ 𝑝 ≤ 1. Consequently

1 ≤ lim ∥𝑢∥2

𝑝 + lim ∥𝑣𝑖 ∥2
𝑝

≤ ∥𝑢∥2

𝑝 + lim

1
𝑠( 𝑛
𝑛−2, 𝜆)
1
lim
𝑠( 𝑛
𝑛−2, 𝜆)

∫

(cid:0)

𝑀

|∇𝑣𝑖 |2 + 𝜆

∫

Σ

(cid:1)

𝑣2
𝑖

∫

𝑀

|∇𝑣𝑖 |2

= ∥𝑢∥2

𝑝 +

𝑣𝑖 can be estimated using (2.6.4),

∫

𝑀

|∇𝑢|2 + 𝜆

∫

Σ

𝑢2 + lim

∫

𝑀

|∇𝑣𝑖 |2 = 𝑠(

𝑛
𝑛 − 2

, 𝜆)

53

where we used 𝑣𝑖 ⇀ 0 ∈ 𝐻1(𝑀) and 𝑣𝑖 → 0 in 𝐿2(Σ). Combine the two equations above and we

get

𝑀 |∇𝑢|2 + ∫
∫
∥𝑢∥2
𝑝

Σ

𝑢2

≤ 𝑠(

𝑛
𝑛 − 2

, 𝜆)

So 𝑢 is a minimizer.

□

Remark 2.6.1. The proof of part ii) comes from [BN83] where the H.Brezis and L.Nirenberg proved

similar results for a different equation

−Δ𝑢 = 𝑢 𝑝 + 𝜆𝑢 on 𝑀

𝑢 > 0 on 𝑀

𝑢 = 0 on Σ

Corollary 2.2. For the unit disk, 𝑠( 𝑛

𝑛−2, 𝑛−2

2 ) admits only constant minimizer for 𝜆 < 𝑛−2

2 , and

admit no minimizer for 𝜆 > 𝑛

𝑛−2.

Proof. It’s well known that 𝑠( 𝑛

𝑛−2, 𝑛−2

2 ) = 𝑛−2

2 |Σ|

1
𝑛−1 . Then the result follows from the two theorems

above and lemma 2.1.1.

□

For the critical power case, 𝑠( 𝑛

𝑛−2 , 𝜆) has a strong relationship with type II Yamabe problem.

Use standard argument and we can get a similar existence theorem

Theorem 2.9. If

4(𝑛−1)
𝑛−2 𝑠( 𝑛

𝑛−2, 𝜆) < 𝑌 (B𝑛, S𝑛−1, 𝑑𝑥2), then it admits a minimizer.

Proof. The trick is again “lowering the index". For each 𝑞 < 𝑛

𝑛−2, 2.2.8 admits a solution 𝑢𝑞
(it might be constant). If 𝑢𝑞’s are uniforma bounded above, then the ellipticity of Dirichlet-to-

Neumann operator implies a universal upper bound for 𝑢𝑞 in 𝐶 𝑘 for any 𝑘 ∈ Z+. Consequently 𝑠𝑞

converges to a solution of (2.2.8) for

bound. Suppose on the contrary that there exits 𝑞𝑘 → 𝑛

𝑛
𝑛−2. So it suffices to show that there doen’t exist such a 𝐿∞
𝑛−2, 𝑢𝑘 and 𝑝𝑘 ∈ Σ so that 𝑢𝑘 minimizes
𝐸𝑞𝑘,𝜆 and 𝑚𝑘 := 𝑢𝑘 ( 𝑝𝑘 ) = sup𝑥∈𝑀 𝑢(𝑥) → ∞. The idea is that we are going to show that by scaling
𝑢𝑘 will “converge" locally around 𝑝 to a solution of 𝑃𝐷𝐸 in upper plane, and this contradicts the

54

assumption 𝑠( 𝑛

𝑛−2, 𝜆) < 𝑌 (B𝑛, S𝑛−1, 𝑑𝑥2). For convenience of readers, I will restate the equations

for 𝑢𝑘

Δ𝑢𝑘 = 0
𝜕𝑢𝑘
𝜕𝜈
𝑢𝑞𝑘+1
𝑘

= 1

= −𝜆𝑢𝑘 + 𝑠𝑢𝑞𝑘
𝑘

∫

Σ

on 𝑀 𝑛

on Σ𝑛−1

(2.6.7)

By compactness we might assume 𝑝𝑘 → 𝑝 ∈ Σ. We might pick local coordinate upper ball

𝑈𝑝 (2𝜖) := B𝑛 (2𝜖) ∩ {𝑥𝑛 ≥ 0} centered at 𝑝, where {𝑥𝑖}1≤𝑖≤𝑛−1 is local normal coordinate for 𝑝 ∈ Σ
and 𝑥𝑛 is the coordinate in normal direction. Let 𝛿𝑘 = 𝑚1−𝑞𝑘
is defined in 𝑈𝑝 ( 𝜖
𝛿𝑘

) for large 𝑖’s with radius 𝜖
𝛿𝑘

→ ∞. By (2.6.7), 𝑣 𝑘 locally satisfies

𝑢𝑘 (𝛿𝑘 𝑥 + 𝑝𝑘 ). Then 𝑣 𝑘

, and 𝑣 𝑘 = 1
𝑚𝑘

𝑘

1
𝑏𝑘

𝜕𝑗 (𝑎𝑖 𝑗

𝑘 𝜕𝑖𝑣 𝑘 ) = 0 on 𝑈𝑝 (

𝜖
𝛿𝑘

)

−

𝜕𝑣 𝑘
𝜕𝑥𝑛

+ 𝑐𝑘 𝑣 𝑘 = 𝑠𝑣𝑞𝑘
𝑘

on 𝑈𝑝 (

𝜖
𝛿𝑘

) ∩ {𝑥𝑛 = 0}

(2.6.8)

where

𝑎𝑖 𝑗
𝑘 (𝑥) = 𝑔𝑖 𝑗 (𝛿𝑘 𝑥 + 𝑝𝑘 ) → 𝛿𝑖 𝑗
𝑏𝑘 (𝑥) = √︁det 𝑔(𝛿𝑘 𝑥 + 𝑝𝑘 ) → 1

𝑐𝑘 = 𝜆𝑚−𝑞𝑘

𝑘 → 0

In the equation above we have an additional − in front of 𝜕𝑣 𝑘
𝜕𝑥𝑛

because 𝜕
𝜕𝑥𝑛

is in the inner normal

direction instead of outer normal direction.

Note that 𝑢𝑘
𝑚𝑘

has uniform 𝐿∞ bound by its definition. Since they satisfy a similar equation

Δ𝑢𝑘 = 0 on 𝑀

𝜕𝑢𝑘
𝜕𝜈

+ 𝑐𝑘𝑢𝑘 = 𝑠𝑢𝑞𝑘
𝑘

onΣ

So 𝑢𝑘 are uniform bounded in any 𝐶𝑘 (𝑀) norm by ellipticity of Dirichlet-to-Neumann operator.

𝑣 𝑘 are defined in 𝑈𝑝 ( 𝜖
𝛿𝑘

𝑘 𝜕𝛼𝑢𝑘 (𝛿𝑘 𝑥 + 𝑝𝑘 ) for 𝛿𝑘 → 0. So 𝑣 𝑘 are also uniformly
bounded in any 𝐶 𝑘 (𝑈𝑝 (𝑅)) for any fixed 𝑅. So we could pick a sub-sequence so that 𝑣𝑖 → 𝑣 so

) and 𝜕𝛼𝑣 𝑘 (𝑥) = 𝛿|𝛼|

that

Δ𝑣 = 0 on H𝑛
+
𝜕𝑣
𝜕𝜈

= 𝑠𝑣 𝑛

𝑛−2 on R𝑛−1

55

(2.6.9)

where 𝜈 is the outer normal direction. (I failed to derive this convergence from (2.6.8) directly

since Dirichlet-to-Neumann operator is global, while 𝑣 𝑘 is only defined locally. Also, the Schauder

estimates can’t be applied directly.)

Now if 𝜈 has enough decay at infinity, we could multiply (2.6.9) by 𝑣 on both sides and integrate to

get

∫

H𝑛
+

|∇𝑣|2 = 𝑠

∫

R𝑛−1

2(𝑛−1)
𝑛−2

𝑣

(2.6.10)

Also, as a limit we probably have ∫
Σ

+ has vanishing mean curva-
ture and scalar curvature. Use 𝑣 as the test function for type II Yamabe problem and we get

2(𝑛−1)
𝑛−2 ≤ 1. Note that H𝑛

𝑣

𝑌 (H𝑛

+, R𝑛−1, 𝑑𝑥2) ≤

4(𝑛−1)
𝑛−2 𝑠. However, it’s well known that the upper half plane and unit disk are

conformally equivalent through

𝐹 (𝑥1, · · · , 𝑥𝑛) =

1 · · · − 𝑥2
𝑛)

(2.6.11)

𝐹 : B𝑛 → H𝑛
1
𝑃

𝐹∗𝑔H𝑛 =

(2𝑥1, · · · , 2𝑥𝑛−1, 1 − 𝑥2
4
𝑃2

𝑔B𝑛 := 𝜙 4

𝑛−2 𝑔B𝑛

where 𝑃 = 𝑥2

1 + · · · + 𝑥2

𝑛−1 + (1 − 𝑥𝑛)2. Use 𝑣𝜙 as test function for B𝑛, and we will get

𝑌 (B𝑛, S𝑛−1, 𝑑𝑥2) = 𝑌 (H𝑛

+, R𝑛−1, 𝑑𝑥2) ≤

4(𝑛 − 1)
𝑛 − 2

𝑠(

𝑛
𝑛 − 2

, 𝜆)

which contradicts to assumption 𝑠( 𝑛

𝑛−2 , 𝜆) < 𝑌 (B𝑛, S𝑛−1, 𝑑𝑥2). Next we will fix the gaps in the

argument above. Let 𝑔′ = 𝑔|Σ. We can compute in local coordinate

∫

𝐵𝑛−1
𝑝

( 𝜖
𝛿𝑘

)

𝑣𝑞𝑘+1
𝑘

√︁det(𝑔′)( 𝑝𝑘 + 𝛿𝑘 𝑥)𝑑𝑥 =

∫

√︁det(𝑔′) (𝑥)
𝑘 𝑚𝑞𝑘+1
𝛿𝑛−1
𝑘
∫

𝐵𝑛−1
𝑝𝑘 (𝜖)
≤ 𝑚 (𝑞−1)(𝑛−1)−𝑞−1

𝑘

𝑢𝑞𝑘+1
𝑘

𝑑𝑦

𝑢𝑞𝑘+1
𝑘

𝑑𝑉𝑔′

Σ

We used change of variable 𝑦 = 𝑝𝑘 + 𝛿𝑘 𝑥 in the first line. Since 𝑞 < 𝑛

𝑛−2, 𝑚 (𝑞−1)(𝑛−1)−𝑞−1

𝑘

≤ 1 for

= 𝑚 ((𝑛−2)𝑞−𝑛
𝑘

56

large 𝑘’s. Since Similarly, we compute

∫

𝑈 𝑝 ( 𝜖
𝛿𝑘

)

𝑎𝑖 𝑗
𝑘 (𝑥)𝜕𝑖𝑣 𝑘 𝜕𝑗 𝑣 𝑘 𝑏𝑘 (𝑥)𝑑𝑥 =

∫

𝑈 𝑝 (

𝜖 )
𝛿 ) 𝑘

𝛿2
𝑘
𝑚2
𝑘
∫

(cid:0)𝑔𝑖 𝑗 𝜕𝑖𝑢𝑘 𝜕𝑗 𝑢𝑘 (cid:1) (𝛿𝑘 𝑥 + 𝑝𝑘 )𝑑𝑥

= 𝑚 (𝑛−2)𝑞−𝑛
𝑘

|∇𝑢𝑘 |2𝑑𝑉𝑔

𝑈 𝑝𝑘 (𝜖)

Since √︁det(𝑔′)( 𝑝𝑘 + 𝛿𝑘 𝑥) → 1 and 𝑎𝑖 𝑗

𝑘 → 𝛿𝑖 𝑗 , and 𝑣 𝑘 converges uniformly in any compact subset,

≤ 𝑚 (𝑛−2)𝑞−𝑛
𝑘

∥∇𝑢𝑘 ∥ 𝐿2 (𝑀) ≤ 𝐶

by Fatou’s lemma we arrive at

∫

R𝑛−1
∫

𝑣

2(𝑛−1)
𝑛−2 ≤ 1

(2.6.12)

|∇𝑣|2 ≤ ∞

H𝑛
+
+, and 𝑣 𝑅 (𝑥) = 𝜂( 𝑥

Let 𝜂(𝑥) be a cut-off function in H𝑛

𝑅 )𝑣. Then with the two bounds above we can

verify that 𝑣 𝑅 → 𝑣 in 𝐻1(H𝑛

+) and 𝐿

2(𝑛−1)
𝑛−2 (R𝑛−1) by showing

∫

H𝑛
+

∫

R𝑛−1

|∇(𝑣 − 𝑣 𝑅)|2 → 0

|𝑣 − 𝑣 𝑅 |

2(𝑛−1)
𝑛−2 → 0

Multiply 𝜂( 𝑥

𝑅 )𝑣 to (2.6.9) and integrate by part, we get

∫

H𝑛
+

⟨∇𝑣, ∇𝑣 𝑅)⟩ = 𝑠

∫

R𝑛−1

𝑣 𝑛
𝑛−2 𝑣 𝑅

Let 𝑅 → ∞, and we get (2.6.10).

□

In view of the previous results, for a fixed Riemannian manifold with boundary we can ask the

following questions:

i) does

4(𝑛 − 1)
𝑛 − 2

𝑠(

𝑛
𝑛 − 2

, 𝜆) → 𝑌 (B𝑛, S𝑛−1, 𝑑𝑥2) as 𝜆 → ∞

ii) does there exists a 𝜆0 so that 𝑠(

, 𝜆0) = 𝑠(

𝑛
𝑛 − 2

, 𝜆) for all 𝜆 > 𝜆0

𝑛
𝑛 − 2
𝑛
𝑛 − 2

iii) does there exists a 𝜆1 so that 𝑠(

, 𝜆) admits only constant minimizer for all 𝜆 < 𝜆1

My guess is all these are correct, but I have not found a way to solve these.

57

CHAPTER 3

A LOG-SOBOLEV INEQUALITY

In this chapter, I will introduce the log-Sobolev inequality, which is closely related to Wang’s

conjecture. The validity of the log-Sobolev inequality provides a key insight into the confidence

we can place in Wang’s conjecture.

3.1 Motivation for Log-Sobolev inequality

Let (𝑀, 𝜕 𝑀 = Σ, 𝑔) a Riemannian manifold with boundary. Wang’s conjecture 2.1 relies on the

boundedness of the trace operator: 𝐻1(𝑀) ↩→ 𝐿𝑞+1(Σ) for 𝑞 ≤ 𝑛

𝑛−2. The embedding is compact
𝑛
𝑛−2, ultimately losing compactness.
Consequently, the conjecture becomes more challenging as 𝑞 increases. For the critical power, the

when the inequality is strict, but it weakens as 𝑞 approaches

existence of the minimizer is uncertain due to the loss of compactness. Due to this, one might

be interested in examining the behavior for small values of 𝑞. If Wang’s conjecture is true, in its

setting we have

∫

𝑀

|∇𝑢|2 + 𝜆

∫

Σ

𝑢2 ≥ |Σ|

𝑞−1
𝑞+1 𝜆(

𝑢𝑞+1)

2
𝑞+1

∫

Σ

for 𝜆(𝑞 − 1) ≤ 1 and 𝑞 ≤ 𝑛

𝑛−2. Let 𝜆 = 1

𝑞−1, and we get

𝑞 − 1
|Σ|

∫

𝑀

|∇𝑢|2 +

1

∫

|Σ|

Σ

𝑢2 ≥ (

1

∫

|Σ|

Σ

𝑢𝑞+1)

2
𝑞+1

Notice that the equality always holds for 𝑞 = 1. Now for an arbitrary 𝑢 ∈ 𝐻1(𝑀) satisfying
∫
Σ

𝑢2 = Σ, take the limit 𝑞 → 1, and we arrive at

∫

2

𝑀

|∇𝑢|2 ≥

∫

Σ

𝑢2 log(𝑢2)

Consider the functional

𝐸 (𝑢) := 2

∫

∫

|∇𝑢|2 −

𝑢2 log(𝑢2),

𝑀

Σ
where 𝑢 ∈ 𝐻1(𝑀),

∫

Σ

𝑢2 = |Σ|

(3.1.1)

(3.1.2)

58

Then 𝐸 (𝑢) is bounded below, which will be proved in the next section.

Its Euler-Lagrangian

equation is

Δ𝑢 = 0
𝜕𝑢
𝜕𝜈

= 𝑢 log 𝑢 + 𝜆𝑢

𝜆 comes from the Lagrangian multiplier. Note that the second equation is not linear, and we could

scale 𝑢 to kill 𝜆𝑢 to get

Δ𝑢 = 0
𝜕𝑢
𝜕𝜈

= 𝑢 log 𝑢

(3.1.3)

Just as Wang’s conjecture, we can pose the following conjecture

Conjecture 3.1. Let (𝑀, 𝜕 𝑀 = Σ, 𝑔) be a compact Riemannian manifold with boundary. Suppose

𝑅𝑖𝑐 ≥ 0 on 𝑀, and 𝐼 𝐼 ≥ 1 on Σ where 𝐼 𝐼 is the second fundamental form, then the following PDE

on 𝑀 𝑛

Δ𝑢 = 0
𝜕𝑢
𝜕𝜈

= 𝑢 log 𝑢

admits no solution other than 𝑢 ≡ 1. Consequently,

∫

2

𝑀

|∇𝑢|2 ≥

∫

Σ

𝑢2 log(𝑢2)

for 𝑢 ∈ 𝐻1(𝑀) and ∫

Σ

𝑢2 = |Σ|.

(3.1.4)

(3.1.5)

3.2 Log-Sobolev Inequality in General Manifold

In this section, I will derive a log-Sobolev inequality for general Riemannian manifolds with

boundary. Although a log-Sobolev inequality can be obtained using Theorem 2.4 and a similar

argument as in the previous section, I will employ a different approach that provides additional

information. These methods are modified from the work of [Rot81a], [Rot81b] and [Rot86], where

O.Rothaus studied log-Sobolev inequality for manifolds without boundary.

59

For arbitrary 𝜌 > 0, consider the functional

𝐸𝜌 (𝑢) := 𝜌

∫

∫

|∇𝑢|2 −

𝑢2 log(𝑢2),

𝑀

Σ
where 𝑢 ∈ 𝐻1(𝑀),

∫

Σ

𝑢2 = |Σ|

(3.2.1)

Lemma 3.2.1. 𝑠𝜌 > −∞ for any 𝜌 > 0, and the infimum can be achieved.

𝑠(𝜌) =

inf
𝑢∈𝐻1 (𝑀)

𝐸𝜌 (𝑢)

Proof. Throughout the proof 𝐶 is a constant independent of and 𝑢 and might change from line to

line. Without loss of generality we might assume 𝑢 > 0. Fix 0 < 𝜖 < 2

𝑢2𝑑𝑆 = 1 and log is a concave function, we mighe use Jensen’s inequality for the measure 𝑢2
|Σ|

∫
Σ
and function 𝑢𝜖 , and we get

𝑛−2. Since we assume
𝑑𝑆

1

∫

|Σ|

Σ

𝑢2 log 𝑢2𝑑𝑆 =

≤

2
𝜖
2
𝜖

∫

log 𝑢𝜖 (cid:0)

Σ
log (cid:0)

∫

Σ

1

Σ

𝑢2
|𝜖 |

𝑑𝑆(cid:1)

𝑢2+𝜖 𝑑𝑆(cid:1)

Use boundedness of trace operator,

2
𝜖

∫

log (cid:0)

Σ

1

Σ

𝑢2+𝜖 𝑑𝑆(cid:1) ≤

log (cid:0)𝐶 ∥𝑢∥2+𝜖

(cid:1)

𝐻1 (𝑀)

2
𝜖
2 + 𝜖
𝜖
≤ 𝜌∥𝑢∥2

≤

log(∥𝑢∥2

𝐻1 (𝑀)) + 𝐶

𝐻1 (𝑀) + 𝐶

(3.2.2)

In the last line we used that 𝛼𝑥 − log 𝑥 is bounded below in 𝑥 for any fixed 𝛼 > 0.

The proof demonstrating the achievability of the infimum is standard. We can pick a minimizing

sequence 𝑢𝑖.

𝑠(𝜌) + 1 ≥ 𝐸𝜌 (𝑢𝑖)

= 𝐸𝜌/2(𝑢𝑖) +

𝜌
2

∫

𝑀

|∇𝑢𝑖 |2 ≥ 𝑠(

𝜌
2

) +

𝜌
2

∫

𝑀

|∇𝑢𝑖 |2

So 𝑢𝑖 are bounded in 𝐻1(𝑀). By Alauoglu theorem we can pick a subsequence that converges
weakly to 𝑢 in 𝐻1(𝑀), and thus ∫

𝑢2
𝑖 log 𝑢𝑖
𝑛−2, and we have |(𝑥2 log 𝑥)′| = |(2 log 𝑥 + 1)𝑥| < 𝐶 (1 + 𝑥1+𝜖 ) for all

𝑀 |∇𝑢𝑖 |2. We only need to check that ∫

converges. Fix 𝜖 < 1

𝑀 |∇𝑢|2 ≤ lim ∫

Σ

60

𝑥 > 0.Therefore

∫

|

Σ

𝑢2
𝑖 log 𝑢𝑖 −

∫

Σ

𝑢2
𝑗 log 𝑢 𝑗 | ≤ 𝐶

≤ 𝐶

∫

Σ
∫

Σ

|𝑢𝑖 − 𝑢 𝑗 | max{1 + 𝑢1+𝜖

𝑖

, 1 + 𝑢1+𝜖

𝑗

}

|𝑢𝑖 − 𝑢 𝑗 | (cid:0)1 + 𝑢1+𝜖

𝑖

+ 1 + 𝑢1+𝜖

𝑗

(cid:1)

≤ ∥𝑢𝑖 − 𝑢 𝑗 ∥ 𝐿2 (Σ) (1 + ∥𝑢𝑖 ∥ 𝐿2+2𝜖 + ∥𝑢 𝑗 ∥ 𝐿2+2𝜖 )

≤ 𝐶 (1 + ∥𝑢𝑖 ∥𝐻1 (𝑀) + ∥𝑢 𝑗 ∥𝐻1 (𝑀)) ∥𝑢𝑖 − 𝑢 𝑗 ∥ 𝐿2 (Σ)

≤ 𝐶 ∥𝑢𝑖 − 𝑢 𝑗 ∥ 𝐿2 (Σ)

□

Now we can look at the Euler-Lagrangian equation for 𝑠(𝜌). By a similar computation as the

previous section, we get

Δ𝑢 = 0 on 𝑀
𝜕𝑢
𝜕𝜈

= 𝜆𝑢 log 𝑢 in Σ where 𝜆 =

2
𝜌

(3.2.3)

For any 𝜌 > 0, the function 𝑠(𝜌) is bounded and increasing with respect to 𝜌. Let 𝑢𝜌 be its
minimizer. It is expected that ∫

𝑀 |∇𝑢𝜌 | decreases to 0 as 𝜌 approaches infinity. There might exist
a critical 𝜌0 such that 𝑠(𝜌0) is achieved by 𝑢 ≡ 1, enabling the establishment of a log-Sobolev

inequality. See section 2 of [Rot81b]. Actually, we have the following stronger theorem.

Theorem 3.1. There exists a 𝜆0 so that for 𝜆 < 𝜆0, (3.2.3) admits no solution other than 𝑢 ≡ 1.

Proof. The proof is similar to theorem 2.4. First we want to bound ∫
we get ∫
Σ

𝑢 log 𝑢 = 0. Then

Σ

𝑢. Integrate (3.2.3) by parts,

∫

𝑢 +

𝑢

Σ∩{𝑢≥𝑒}

𝑢 log 𝑢

(3.2.4)

∫

∫

Σ

𝑢 =

Σ∩{𝑢<𝑒}
∫

≤ 𝑒|Σ| +

= 𝑒|Σ| +

Σ∩{𝑢≥𝑒}

∫

Σ∩{𝑢≤𝑒}

𝑢 log 𝑢 ≤ 𝐶1

61

Let 𝐾 (𝑥, 𝑦) be the Schwarz kernel for Dirichlet-to-Neumann operator. From section 3.2 we know

that for 𝑝 < 𝑛−1

𝑛−2, ∥𝐾 (𝑥, ·)∥ 𝐿 𝑝 (Σ) ≤ 𝐶 for all 𝑥 ∈ Σ. Let 0 < 𝑡 < 1, 𝑝∗ =

𝑝
𝑝−1 > 1 and 𝑀 = sup𝑥∈Σ

𝑢

|𝑢(𝑥) −

1

∫

|Σ|

Σ

𝑢| = 𝜆|

∫

Σ

𝐾 (𝑥, 𝑦)𝑢(𝑦) log 𝑢(𝑦)𝑑𝑦|

≤ 𝐶𝜆∥𝑢 log 𝑢∥ 𝑝∗

≤ 𝐶𝜆𝑀 𝑡 ∥𝑢1−𝑡 log 𝑢∥ 𝑝∗

Apparently, there exists a constant 𝐶2( 𝑝∗, 𝑡) such that 𝑥1−𝑡 𝑝 ∗ (log 𝑥) 𝑝∗ ≤ 𝐶2 + 𝑥 for all 𝑥 > 0

provided that (1 − 𝑡) 𝑝∗ < 1, which is achievable. At maximal point, we have

𝑀 − 𝐶1 ≤ |𝑢(𝑥) −

1

∫

𝑢|

Σ

|Σ|
∫

≤ 𝐶𝜆𝑀 𝑡 (

𝐶2 + 𝑢) ≤ 𝐶𝜆𝑀 𝑡

Therefore 𝑢 is bounded provided 𝜆 is bounded above. Then use lemma 2.2.4, we have

Σ

∫

|∇𝑢|2 ≤ 𝐶

∫

𝑀

𝜕𝑢
𝜕𝜈

|

|2

𝑢 log 𝑢

Σ
∫

Σ
∫

Σ

𝑢

𝜕𝑢
𝜕𝜈

𝜕𝑢
𝜕𝜈

= 𝐶𝜆

≤ 𝐶𝜆

= 𝐶𝜆

∫

𝑀

|∇𝑢|2

Note constant 𝐶 is independant of 𝜆 as long as 𝜆 is bounded above. Therefore ∫

small 𝜆, thus 𝑢 ≡ 1.

𝑀 |∇𝑢|2 = 0 for
□

3.3 Flow Method for Manifolds without Boundary

It’s well known that on manifolds without boundary we can solve 𝑢𝑡 = Δ𝑢 and 𝑢 converges to
∫ 𝑢0. If we run this flow and keep track of how ∫ |∇𝑢|2 decreases in 𝑡, hopefully

the constant 1
|Σ|

we can get something. Actually, this idea works for Gaussian measure 𝑑𝜇 =
It’s well known that ∫

R𝑛 𝑑𝜇 = 1, i.e. 𝑑𝜇 is a probability on R𝑛. Define
∫

E( 𝑓 ) :=

𝑓 𝑑𝜇

R𝑛
ˆΔ 𝑓 := Δ 𝑓 − ⟨𝑥, ∇ 𝑓 ⟩

1

(2𝜋)𝑛/2 𝑒− | 𝑥 |

2
2 𝑑𝑥 on R𝑛.

𝑢(𝑡, 𝑥) = 𝑃𝑡 𝑓 := E𝜁 (cid:0) 𝑓 (𝑒−𝑡𝑥 +

√︁1 − 𝑒−2𝑡 𝜁)(cid:1)

62

Then 𝑢 defined as above solves 𝑢𝑡 = ˆΔ𝑢. Using this flow we can show that

Theorem 3.2. If 𝑓 is 𝐶1(R𝑛), E( 𝑓 ) = 1 and E(|∇ 𝑓 |)2 ≤ ∞, then

E( 𝑓 2 log 𝑓 2) ≤ 2E(|∇ 𝑓 |2)

This method can be carried to Riemannian manifolds as follows

Theorem 3.3. Let (𝑀, 𝑔) be a compact Riemannian manifold without boundary. Suppose 𝑅𝑖𝑐 ≥
(𝑛 − 1)𝑔, then for all 𝑓 ∈ 𝐻1(𝑀), 𝑓 > 0 and ∫

𝑓 = |𝑀 |, we have

𝑀

1
2(𝑛 − 1)

∫ |∇ 𝑓 |2

𝑓

∫

≥

𝑓 log 𝑓

If we pick 𝑓 2 in the inequality, we get

∫

1
𝑛 − 1

∫

|∇ 𝑓 |2 ≥

𝑓 2 log 𝑓

Proof. Let 𝑢 be solutions of

𝑢𝑡 = Δ𝑢

𝑢(0, ·) = 𝑓

(3.3.1)

Then

Note that 𝑢 →

𝜕
𝜕𝑡
∫

(𝑢 log 𝑢) = (log 𝑢 + 1)𝑢𝑡 = (log 𝑢 + 1)Δ𝑢
|∇𝑢|2
𝑢

𝑢 log 𝑢 = −

∫

𝑀

𝑀

𝜕
𝜕𝑡

∫

𝑓

𝑀

|𝑀 | = 1 in 𝐻1(𝑀). Integrate in time,

∫ ∞

∫

0

𝑀

|∇𝑢|2
𝑢

𝑑𝑉 𝑑𝑡 =

∫

𝑀

𝑓 log 𝑓

(3.3.2)

Use Bochner’s formula, we compute

𝜕
𝜕𝑡

|∇|2 = 2⟨∇𝑢𝑡, ∇𝑢⟩ = 2⟨∇Δ𝑢, ∇𝑢⟩

= Δ|∇𝑢|2 − 2|∇2𝑢|2 − 2𝑅𝑖𝑐(∇𝑢, ∇𝑢)

(3.3.3)

≤ Δ|∇𝑢|2 − 2|∇2𝑢|2 − 2(𝑛 − 1)|∇𝑢|2

63

Let 𝑣 =

√

𝑢, then

4

𝜕
𝜕𝑡

|∇𝑣|2 =

𝜕
𝜕𝑡

|∇𝑢2|
𝑢

1
𝑢

𝜕|∇𝑢|2
𝜕𝑡

Δ𝑢|∇𝑢|2
𝑢2

−

=

4Δ|∇𝑣|2 = Δ

|∇𝑢|2
𝑢
Δ|∇𝑢|2
𝑢
Δ|∇𝑢|2
𝑢

=

=

− 2

− 2

⟨∇|∇𝑢|2, ∇⟩
𝑢2
⟨∇|∇𝑢|2, ∇⟩
𝑢2

1
𝑢

+ |∇𝑢|2Δ
|∇𝑢|4
𝑢3 −

+ 2

Δ𝑢|∇𝑢|2
𝑢2

Take difference between the two equations above and use (3.3.3),

4(

𝜕
𝜕𝑡

− Δ)|∇𝑣|2 ≤ −

2(𝑛 − 1)|∇𝑢|2
𝑢
2(𝑛 − 1)|∇𝑢|2
𝑢

= −

= −

2(𝑛 − 1)|∇𝑢|2
𝑢

≤ −

2(𝑛 − 1)|∇𝑢|2
𝑢

2|∇2𝑢|2
𝑢

−

+ 2

−

−

2
𝑢

2
𝑢

∑︁

1≤𝑖, 𝑗 ≤𝑛

∑︁

1≤𝑖, 𝑗 ≤𝑛

⟨∇|∇𝑢|2, ∇⟩
𝑢2
𝑢𝑖 𝑗 𝑢𝑖𝑢 𝑗
𝑢

𝑖 𝑗 − 2

(cid:0)𝑢2

|𝑢𝑖 𝑗 −

𝑢𝑖𝑢 𝑗
𝑢

|2

Integrate this inequality in both space and time,

∫ ∞

∫

0

𝑀

2(𝑛 − 1)|∇𝑢|2
𝑢

−

≥

=

=

− Δ)|∇𝑣|2𝑑𝑉 𝑑𝑡

4(

𝜕
𝜕𝑡
𝜕
𝜕𝑡
∫

4

∫ ∞

∫

𝑀

0
∫ ∞

∫

0
∫ ∞

0

(cid:0)

𝑀
𝜕
𝜕𝑡
∫

= lim
𝑡→∞
∫

= −

𝑀

𝑀
|∇ 𝑓 |2
𝑓

𝑑𝑉

|∇𝑣|2𝑑𝑉 𝑑𝑡

4|∇𝑣|2𝑑𝑉 (cid:1) 𝑑𝑡

𝑀

4|∇𝑣(𝑡, ·)|2𝑑𝑉 −

∫

𝑀

4|∇𝑣(0, ·)|2𝑑𝑉

− 2

|∇𝑢|4
𝑢3
𝑖 𝑢2
𝑢2
𝑗
𝑢2

(cid:1)

+

(3.3.4)

(3.3.5)

Combine (3.3.3) and (3.3.5), we get desired result.

□

3.4 Sectional Curvature Results

Cut(Σ), which is a closed set in the interior of 𝑀 and is of measure zero. Consider 𝜓 := 𝜌2 −

Let 𝜌(𝑥) = 𝑑 (𝑥, 𝜎) be the distance from the boundary. It’s smooth away from the cut locus
𝜌2
2 .
If (𝑀, Σ, 𝑔) is assumed to have non-negative sectional curvature and 𝐼 𝐼 ≥ 1, then by the Hessian

64

comparison theorem (cf. [Kas82]),

−∇2𝜙 ≥ 𝑔

Furthermore, 𝜓 has nice property near the boundary

𝜓Σ = 0
𝜕𝜓
𝜕𝜈

= −1

These prove advantageous when we use ∇𝜙 as the testing field in (2.4.7). But the problem is cut

locus. To overcome this difficulty, in [XX19] the C.Xia and C.Xiong has the following construction.

Theorem 3.4. Suppose (𝑀, Σ, 𝑔) has non-negative sectional curvature and 𝐼 𝐼 ≥ 1. Fix a neigh-

borhood C of Cut(Σ) in the interior of 𝑀. Then for any 𝜖 > 0, there exists a smooth non-negative

function 𝜓𝜖 on 𝑀 such that 𝜓𝜖 = 𝜙 on 𝑀 \ C and

−∇2𝜓𝜖 ≥ (1 − 𝜖)𝑔

(3.4.1)

In [GHW19], the authors use this function in Wang’s conjecture and get the following

Theorem 3.5 (Q.Guo, F.Hang and X.Wang). Let (𝑀, Σ, 𝑔) be as in Wang’s conjecture. Then the
only positive solutions to (2.1.1) is constant if (𝑞 − 1)𝜆 ≤ 1 provided 2 ≤ 𝑛 ≤ 8 and 1 < 𝑞 ≤ 4𝑛

5𝑛−9.

Consequently,

for these (𝑞, 𝜆).

𝑞 − 1
|Σ|

∫

𝑀

|∇𝑢|2 +

1

∫

|Σ|

Σ

𝑢2 ≥ (

1

∫

|Σ|

Σ

𝑢𝑞+1)

2
𝑞+1

(3.4.2)

Their method also works for (3.1.3).

Theorem 3.6. Let (𝑀, Σ, 𝑔) be as in Wang’s conjecture. Suppose 2 ≤ 𝑛 ≤ 8 Then the only positive

solution to (3.1.3) is 𝑢 ≡ 1.

Proof. Let 𝑢 be a solution to (3.1.3). Let 𝑎, 𝑏 bw two constants that will be determined later. Set

𝑢 = 𝑣−𝑎, then

|∇𝑣|2
𝑣

Δ𝑣 = (𝑎 + 1)

𝜕𝑣
𝜕𝜈

= 𝑣 log 𝑣

65

(3.4.3)

We have the following two lemmas from [GHW19]:

Lemma 3.4.1. Suppose 𝜙|Σ = 0 and

𝜕𝜙
𝜕𝜈 = −1, then for any smooth 𝑣 and 𝑏 ∈ R

∫

(1 −

1
𝑛

)(Δ𝑣)2𝑣𝑏𝜙 +

𝑏
2

𝜙𝑣𝑏−2|∇𝑣|2 (cid:0)3𝑣Δ𝑣 + (𝑏 − 1)|∇𝑣|2(cid:1)

𝑣𝑏∇2𝜙(∇𝑣, ∇𝑣) − |∇𝑣|2𝑣𝑏Δ𝜙 −

𝑏
2

|∇𝑣|2𝑣𝑏−1⟨∇𝑣, ∇𝜙⟩

(3.4.4)

𝑀
∫

=

𝑀
+ (cid:0)|∇2𝑣 −

Δ𝑣
𝑛

𝑔|2 + 𝑅𝑖𝑐(∇𝑣, ∇𝑣)(cid:1)𝑣𝑏𝜙 −

∫

Σ

𝑣𝑏 |∇Σ𝑣|2

Lemma 3.4.2. The proof of the first lemma is similar to that of usual Reilly’s formula, and the

proof of the second one is based on Pohozaev identity (2.4.7). Under the same assumptions as in

lemma 3.4.1, we have

∫

𝑀
1
2

=

𝑣𝑏∇2𝜙(∇𝑣, ∇𝑣) + (𝑣Δ𝑣 +

∫

Σ

𝑣𝑏 (|∇Σ𝑣|2 − (

𝜕𝑣
𝜕𝜈

)2)

𝑏
2

|∇𝑣|2)𝑣𝑏−1⟨∇𝑣, ∇𝜙⟩ −

1
2

𝑣𝑏 |∇𝑣|2Δ𝜙

Apply these two lemmas for 𝑣 in (3.4.3), we get respectively

𝑄 :=

= (cid:0)(1 −

Δ𝑣
𝑛
)(𝑎 + 1)2 +

(cid:0)|∇2𝑣 −
1
𝑛

𝑔|2 + 𝑅𝑖𝑐(∇𝑣, ∇𝑣)(cid:1)𝑣𝑏𝜙
𝑏(3𝑎 + 𝑏 + 2)
2

∫

(cid:1)

−𝑣𝑏∇2𝜙(∇𝑣, ∇𝑣) + |∇𝑣|2𝑣𝑏Δ𝜙 +

𝑣𝑏−2|∇𝑣|4𝜙

𝑀
𝑏
2

|∇𝑣|2𝑣𝑏−1⟨∇𝑣, ∇𝜙⟩ +

∫

Σ

𝑣𝑏 |∇Σ𝑣|2

∫

+

𝑀

and

∫

𝑣𝑏∇2𝜙(∇𝑣, ∇𝑣) + (cid:0)𝑎 + 1 +

𝑏
2

(cid:1)∇𝑣|2𝑣𝑏−1⟨∇𝑣, ∇𝜙⟩ −

1
2

𝑣𝑏 |∇𝑣|2Δ𝜙

𝑀
1
2

∫

𝜕𝑣
𝜕𝜈
Combine these two equalities to eliminate terms involving ⟨∇𝑣, ∇𝜙⟩, we get

𝑣𝑏 (|∇Σ𝑣|2 − (

)2)

=

Σ

(3.4.5)

(3.4.6)

(3.4.7)

𝑄 =(cid:0)(1 −

1
𝑛

)(𝑎 + 1)2 +

+

+

∫

𝑀

∫

Σ

−

−

𝑎 + 1 + 𝑏
𝑎 + 1 + 𝑏/2
𝑏/4
𝑎 + 1 + 𝑏/2

𝑣𝑏 (

∫

(cid:1)

𝑣𝑏−2|∇𝑣|4𝜙

𝑏(3𝑎 + 𝑏 + 2)
2

𝑀
𝑣𝑏∇2𝜙⟨∇𝑣, ∇𝑣⟩ +

𝑎 + 1 + 3𝑏/4
𝑎 + 1 + 𝑏/2

|∇𝑣|2𝑣𝑏Δ𝜙

𝜕𝑣
𝜕𝜈

)2 +

𝑎 + 1 + 3𝑏/4
𝑎 + 1 + 𝑏/2

𝑣𝑏 |∇Σ𝑣|2

66

Set 𝑎 + 1 + 3𝑏

4 = 0 to eliminate terms involving Δ𝜙 and |∇Σ𝑣|2, and take 𝜙 to be 𝜙𝜖 as in theorem

3.4, we get

where

𝑄𝜖 ≤ (cid:0) (5𝑛 − 9 − (𝑛 + 9)𝑎) (𝑎 + 1)
∫

9𝑛

∫

− (1 − 𝜖)

𝑣𝑏 |∇𝑣|2 +

∫

(cid:1)

𝑣𝑏−2|∇𝑣|4𝜓𝜖

𝑀
𝑣𝑏∇2𝜓(∇𝑣, ∇𝑣) +

𝑀\C

∫

Σ

𝑣𝑏 (

𝜕𝑣
𝜕𝜈

)2

C

∫

𝑀

𝑄𝜖 :=

(cid:0)|∇2𝑣 −

Δ𝑣
𝑛

𝑔|2 + 𝑅𝑖𝑐(∇𝑣, ∇𝑣)(cid:1)𝑣𝑏𝜓𝜖

Now let 𝜖 → 0 and then shrink C. Notice that Δ𝜙 ≤ −𝑔 whenever its smooth. It yields

𝑄 ≤ (cid:0) (5𝑛 − 9 − (𝑛 + 9)𝑎) (𝑎 + 1)
𝜕𝑣
𝜕𝜈

𝑣𝑏 |∇𝑣|2 +

𝑣𝑏 (

9𝑛

−

∫

∫

)2

𝑀

Σ

∫

(cid:1)

𝑀

𝑣𝑏−2|∇𝑣|4𝜓𝜖

(3.4.8)

where

𝑄 :=

∫

𝑀

(cid:0)|∇2𝑣 −

Δ𝑣
𝑛

𝑔|2 + 𝑅𝑖𝑐(∇𝑣, ∇𝑣)(cid:1)𝑣𝑏𝜓

Compute ∫
Σ

𝑣𝑏 ( 𝜕𝑣

𝜕𝜈 )2 as follows

= 𝜆

∫

𝑀

∫

Σ

𝑣𝑏 (

𝜕𝑣
𝜕𝜈

)2 = 𝜆

∫

Σ

𝑣𝑏+1 log 𝑣

𝜕𝑣
𝜕𝜈

𝑣𝑏+1 log 𝑣Δ𝑣 + (𝑏 + 1)𝑣𝑏 log 𝑣|∇𝑣|2 + 𝑣𝑏 |∇𝑣|2

= 𝜆

∫

𝑀

(𝑎 + 𝑏 − 2)𝑣𝑏 log 𝑣|∇𝑣|2 + 𝑣𝑏 |∇𝑣|2

Plug this equality in (3.3.5),

𝑄 ≤ (cid:0) (5𝑛 − 9 − (𝑛 + 9)𝑎) (𝑎 + 1)
∫

9𝑛

∫

(cid:1)

𝑀

+ (𝜆 − 1)

𝑣𝑏 |∇𝑣|2 + 𝜆(𝑎 + 𝑏 − 2)

𝑣𝑏−2|∇𝑣|4𝜓𝜖

𝑀

∫

𝑀

𝑣𝑏 log 𝑣|∇𝑣|2

We want 𝑎+𝑏−2 = 0 since we don’t know the sign for ∫
𝑀
we get 𝑎 = 2, 𝑏 = −4. Additionally, we aim for (5𝑛−9−(𝑛+9)𝑎)(𝑎+1)

𝑣𝑏 log 𝑣|∇𝑣|2. Together with 𝑎+1+ 3𝑏

4 = 0,
, which imposes the condition

9𝑛

𝑛 ≤ 9. This completes our theorem.

□

Corollary 3.1. Under the same assumptions,

∫

2

𝑀

|∇𝑢|2 ≥

∫

Σ

𝑢2 log(𝑢2)

67

for 𝑢 ∈ 𝐻1(𝑀) and ∫

Σ

𝑢2 = |Σ|

Remark 3.4.1. If we take derivative with respect to 𝑞 at 𝑞 = 1 for (3.4.2), just as we did in section

4.1, we get the desired log-Sobolev inequality (3.1.1). But using their method, we also proves

non-existence of non-constant solutions to (3.1.3), which is stronger.

In [GHW19], the authors applied maximal principle for 𝑛 = 2 and proved the following

Theorem 3.7 (Q.Guo, F.Hang and X.Wang). Let (𝑀, Σ, 𝑔) be as in Wang’s conjecture and 𝑛 = 2.

Then the only positive solutions to (2.1.1) is constant if (𝑞 − 1)𝜆 ≤ 1 provided 𝑞 ≥ 2.

This maximal principle also works for our case. But since it’s fully covered by the previous

result, I won’t include it here.

3.5 Ricci Curvature Results

An obstacle in both Wang’s Conjecture 2.1 and Conjecture 3.1 is the lack of a comprehensive

understanding of how Ricci curvature affects the Dirichlet-to-Neumann operator. Although some

partial results have been obtained under the assumption of sectional curvature ≥ 0, as discussed in

the previous section and presented in [GHW19], no progress has been made under the condition

𝑅𝑖𝑐 ≥ 0. In this section, I will present a result in this direction.

Theorem 3.8. (𝑀 𝑛, Σ, 𝑔) a Riemannian manifold with boundary. Suppose 𝑛 ≤ 8, 𝑅𝑖𝑐 ≥ 0 on 𝑀,

II ≥ 𝑔Σ and 𝑅𝑖𝑐Σ ≥ (𝑛−2)𝑔Σ on Σ, then there exists a 𝜆0 that only depends on the dimension so that

on 𝑀 𝑛

= 𝜆𝑢 log 𝑢 on Σ𝑛−1

Δ𝑢 = 0
𝜕𝑢
𝜕𝜈

∫

Σ

𝑢2dS ≤ 𝐴(Σ)

(3.5.1)

admit no non-constant solution provided 𝜆 ≤ 𝜆0

68

Proof. Let 𝑢 = 𝑣−𝛽, then 𝑣 satisfy the following

Δ𝑣 = (1 + 𝛽)

|∇𝑣|2
𝑣

on 𝑀 𝑛

𝜕𝑣
𝜕𝜈

= 𝜆𝑣 log 𝑣

on Σ𝑛−1

(3.5.2)

Define 𝐸𝑖 𝑗 = 𝑣𝑖 𝑗 − Δ𝑣

𝑛 𝛿𝑖 𝑗 and 𝐿𝑖 𝑗 =

𝑣 − |∇𝑣|2
𝑣𝑖𝑣 𝑗

𝑛𝑣 𝛿𝑖 𝑗 . From the work of [LO23], we have

(𝑣𝑎𝐸𝑖 𝑗 𝑣𝑖) 𝑗 ≥ 𝑣𝑎 [𝐸𝑖 𝑗 +

𝑎 + 2(𝛽 + 1) 𝑛−1
𝑛
2

𝐿𝑖 𝑗 ]2 + 𝑐𝑣𝑎−2|∇𝑣|4 := 𝑄

(3.5.3)

where

𝑐 =

𝑛 − 1
𝑛

(𝛽 + 1)

2 + 2𝛽 − 𝑛
𝑛

−

𝑛 − 1
4𝑛

[𝑎 + 2(𝛽 + 1)

𝑛 − 1
𝑛

]2

(3.5.4)

In [LO23] Ou and Lin work on the unit ball. The calculation is essentially the same, and the only

difference is that we used Bochner formula and finally get an inequality. Pick a frame {𝑒}1≤𝛼≤𝑛−1

along the boundary, and let 𝑒𝑛 = 𝜈 be the outer normal. Integrate by parts, and we have

∫

𝑀

div(cid:0)𝑣𝑎𝐸 (∇𝑣, ·)(cid:1)dV =

=

∫

Σ
∫

Σ

𝑣𝑎𝐸 (∇𝑣, 𝜈)dS

𝑣𝑎𝐸 (∇Σ𝑣, 𝜈)dS +

∫

Σ

We calculate 𝐴 and 𝐵 as follows.

𝑣𝑎𝐸 (𝑣𝑛𝜈, 𝜈)dS = 𝐴 + 𝐵

(3.5.5)

𝐴 =

=

≤

∫

Σ
∫

Σ
∫

Σ

𝑣𝑎∇2𝑣(∇Σ𝑣, 𝜈)dS

𝑣𝑎 ⟨∇Σ𝑣, ∇Σ𝑣𝑛⟩ − 𝑣𝑎II(∇Σ𝑣, ∇Σ𝑣)dS

(3.5.6)

𝜆𝑣𝑎 log 𝑣|∇Σ𝑣|2 + (𝜆 − 1)𝑣𝑎 |∇Σ𝑣|2dS

As for 𝐵, we have Δ = ΔΣ + ( 𝜕

𝜕𝜈 )2 + 𝐻 𝜕

𝜕𝜈 on Σ. Using (3.5.2), 𝐵 can be calculated as

69

∫

Σ
∫

Σ
∫

Σ
∫

𝐵 =

=

=

≤

Σ

)dS

𝑣𝑎𝑣𝑛

𝑣𝑎𝑣𝑛 (𝑣𝑛𝑛 −

Δ𝑣
𝑛
(cid:16) (𝑛 − 1)(1 + 𝛽)
𝑛
(𝑛 − 1)(1 + 𝛽)
𝑛
(𝑛 − 1)(1 + 𝛽)
𝑛

|∇𝑣|2
𝑣

− ΔΣ𝑣 − 𝐻𝑣𝑛

(cid:17)

𝑣𝑎−1𝑣𝑛 (|∇Σ𝑣|2 + 𝑣2

𝑛) − 𝐻𝑣𝑎𝑣2

𝑛 − 𝑣𝑎𝑣𝑛ΔΣ𝑣dS

𝑣𝑎−1𝑣𝑛 (|∇Σ𝑣|2 + 𝑣2

𝑛) − (𝑛 − 1)𝑣𝑎𝑣2

𝑛 − 𝑣𝑎𝑣𝑛ΔΣ𝑣dS

(3.5.7)

∫

=

(𝛽 + 1)(𝑛 − 1)
𝑛

𝜆(𝑎 + 1 +
+ 𝜆3 (𝛽 + 1)(𝑛 − 1)
Combine (3.5.5), (3.5.6) and (3.5.7), we have

𝑛

Σ

𝑣𝑎+2 log3 𝑣 + 𝜆𝑣𝑎 |∇Σ𝑣|2dS

)𝑣𝑎 log 𝑣|∇Σ𝑣|2 − (𝑛 − 1)𝜆2𝑣𝑎+2 log2 𝑣

∫

𝑄 ≤

(𝛽 + 1)(𝑛 − 1)
𝑛

Σ

𝜆(𝑎 + 2 +
+ 𝜆3 (𝛽 + 1)(𝑛 − 1)

𝑛

𝑣𝑎+2 log3 𝑣 + (2𝜆 − 1)𝑣𝑎 |∇Σ𝑣|2dS

)𝑣𝑎 log 𝑣|∇Σ𝑣|2 − (𝑛 − 1)𝜆2𝑣𝑎+2 log2 𝑣

(3.5.8)

Set 𝑥 = (1+𝛽)(𝑛−1)

𝑛

It becomes

, and 𝑎 = −2 − 𝑥 to kill the first term on the right hand side of the above inequality.

𝑄 ≤

∫

Σ

−(𝑛 − 1)𝜆2𝑣−𝑥 log2 𝑣 + 𝜆3𝑥𝑣−𝑥 log3 𝑣 + (2𝜆 − 1)𝑣−𝑥−2|∇Σ𝑣|2dS

(3.5.9)

We further require 𝑃 ≥ 0 to make sure 𝑄 ≥ 0. (3.5.4) becomes

𝑃(𝑥) = −

𝑛2 − 10𝑛 + 1
4𝑛(𝑛 − 1)

𝑥2 −

1
𝑛

𝑥 −

𝑛 − 1
𝑛

(3.5.10)

Next we estimate the middle term in (3.5.9) and lower the power for log3 𝑣.

∫

𝜆3𝑥

Σ

𝑣−𝑥 log3 𝑣dS = 𝜆2𝑥

= 𝜆2𝑥

∫

Σ
∫

𝑀

𝑣−𝑥−1 log2 𝑣𝑣𝑛dS

(cid:16)

(−𝑥 − 1) log2 𝑣 + 2 log 𝑣

(cid:17)

𝑣−𝑥−1|∇𝑀 𝑣|2dV

We pick 𝑥 > 0 so that −𝑥 − 1 < 0. Using (−𝑥 − 1) log2 𝑣 + 2 log 𝑣 ≤ 𝑐 for constant 𝑐 > 0. Note that

𝑎 only depends on dimension, and therefore 𝑐. So the above equality becomes

70

∫

𝜆3𝑥

Σ

𝑣−𝑥 log3 𝑣dS ≤ 𝜆2𝑐𝑥

∫

𝑣−𝑥−2|∇𝑀 𝑣|2

𝑀
∫

𝑐𝑥𝜆3
−𝑥 + 𝛽
𝑐(𝑛 − 1)𝑥𝜆3
𝑥 − 𝑛 − 1

Σ

=

=

∫

Σ

𝑣−𝑥 log 𝑣dS

𝑣−𝑥 log 𝑣

(3.5.11)

The second equality can be derived if we multiply (3.5.2) by 𝑣−𝑥−1 and integrate by parts, which

gives 𝜆 ∫
Σ

𝑣−𝑥 log 𝑣 = (−𝑥 + 𝛽) ∫
𝑀

𝑣−𝑥−2|∇𝑀 𝑣|2. Let 𝑤 = 𝑣−𝑥/2. Then (3.5.9) and (3.5.11) give us

∫

𝑄 ≤

4
𝑥2
Finally, we want to bound ∫
Σ

Σ

(2𝜆 − 1)|∇Σ𝑤|2 − (𝑛 − 1)𝜆2𝑤2 log2 𝑤 −

𝑐𝑥2𝜆3
2(𝑥 − 𝑛 − 1)

𝑤2 log 𝑤dS

(3.5.12)

𝑤2 log 𝑤dS by ∫
Σ

|∇Σ𝑤|2dS using theorem 3.3 since the coefficient
for 𝑤2 log 𝑤 is positive and 𝑅𝑖𝑐Σ ≥ 𝑔Σ by our assumption, where 𝑅𝑖𝑐Σ is Ricci curvature on Σ.
∫
𝑤2
2(𝑛−1)
𝛽 , and 𝑥
𝛽 < 2 provided 𝑥 ≥
𝑛+1 . By
Σ
𝐴(Σ)
∫
𝑤2
𝐴(Σ) ≤ 1, and therefore the tail
Σ

Before that, let us figure out the sign for log

and our assumption that ∫
Σ
term in theorem 3.3 could be ignored. Now (3.5.12) is

Hölder inequality for

𝑢2 ≤ 𝐴(Σ),

. 𝑤2 = 𝑢

dS
𝐴(Σ)

𝑥

∫

𝑄 ≤

4
𝑥2

𝑐(𝑛 − 1)𝑥2𝜆3
2(𝑥 − 𝑛 − 1)
𝑛+1 ≤ 𝑥 < 𝑛 − 1 and 𝑃(𝑥) ≥ 0. If we put 𝑥 = 𝑛 − 1, 𝑃(𝑛 − 1) = (9−𝑛)(𝑛−1)2

)|∇Σ𝑤|2 − (𝑛 − 1)𝜆2𝑤2 log2 𝑤

(2𝜆 − 1 −

4𝑛

Σ

where 2(𝑛−1)

(3.5.13)

, so admissible

𝑥 could be found provided 2 ≤ 𝑛 ≤ 8. After picking such a 𝑥 that only depends on dimension, 𝑐

in (3.5.11) is also determined. If 𝜆 > 0 is small enough, we have from (3.5.13) that 0 ≤ 0 and

therefore 𝑢 ≡ 1.

Corollary 3.2. Let (𝑀, Σ, 𝑔) as in theorem 3.8, then

∫

𝑀

|∇𝑢|2dV ≥ 𝜆0

∫

Σ

𝑢2 log 𝑢dS

for ∫

Σ

𝑢2dS = 𝐴(Σ).

□

(3.5.14)

71

Proof. For each 𝜆 > 0, we can show that

∫

𝑀

|∇𝑢|2dV − 𝜆

∫

Σ

𝑢2 log 𝑢dS

(3.5.15)

is bounded below for ∫
Σ

𝑢2 = 𝐴(Σ) and the infimum can be achieved. Let 𝑎𝜆 be the infimum and 𝑢

be the minimizer. From the Euler-Lagrangian equation, we have

Δ𝑢 = 0 on 𝑀
𝜕𝑢
𝜕𝜈

= 𝜆𝑢(log 𝑢 + 𝑎𝜆)

on Σ

𝑢2dS = 𝐴(Σ)

∫

Σ

We can scale to get rid of 𝑎𝜆, namely take 𝑣 = 𝑒𝑎𝜆𝑢 ≤ 𝑢. The equation for 𝑣 is

Δ𝑣 = 0 on 𝑀
𝜕𝑣
𝜕𝜈
𝑣2dS ≤

= 𝜆𝑣 log 𝑣

∫

∫

on Σ

𝑢2dS = 𝐴(Σ)

Σ
The last inequality holds because as the infimum, 𝑎𝜆 ≤ 0. For 𝜆 ≤ 𝜆0, 𝑣 is constant by theorem

Σ

3.8, and so is 𝑢. Therefore 𝑎𝜆 = 0 and the proof is done.

□

72

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