BUBBLE TREE CONSTRUCTION FOR HARMONIC MAPS USING DELIGNE-MUMFORD MODULI SPACE By Woongbae Park A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics - Doctor of Philosophy 2021 ABSTRACT BUBBLE TREE CONSTRUCTION FOR HARMONIC MAPS USING DELIGNE-MUMFORD MODULI SPACE By Woongbae Park We formulate and prove a general compactness theorem for harmonic maps. Convergence is defined using Deligne-Mumford space and families of curves. Given a sequence of harmonic maps from a sequence of closed Riemann surfaces to a compact Riemannian manifold with uniformly bounded energy, the main theorem shows that there is a family of curves and a subsequence such that both the domains and the maps converge off the set of “non-regular” nodes. This convergence result extends existing bubble-tree construction to the case of varying domains. To my parents, Hye-sook Park, Byong-ik Park, and my beloved wife, Younkyung Choi. iii ACKNOWLEDGMENTS First of all and foremost, I would like to thank my advisor, Dr. Thomas H. Parker for his valuable advices and comments. Without his insightful suggestions, my dissertation cannot be completed. I also would like to thank Dr. Russell Schwab, my temporary advisor, for his warm advices and many helps. I appreciate my committee members, Dr. Xiaodong Wang, Dr. Benjamin Schmidt and Dr. Thomas Walpuski, for their time, efforts and valuable suggestions. iv TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background on Harmonic Maps . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Uhlenbeck Compactness Theorem and Bubble-tree extension . . . . . . . . . 3 Chapter 2 Convergence in changing domain . . . . . . . . . . . . . . . . . . 7 2.1 Families of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Deligne-Mumford moduli space . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Sequences of Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Chapter 3 Neck analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1 General neck analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Regular node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 non-regular node I - counterexample . . . . . . . . . . . . . . . . . . . . . . 28 3.4 non-regular node II - monotonicity condition . . . . . . . . . . . . . . . . . . 31 Chapter 4 Adding marked points to build new family . . . . . . . . . . . . 37 4.1 New family and forgetful map . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Coordinate expression for Smooth case (1) . . . . . . . . . . . . . . . . . . . 41 4.3 Coordinate expression for Nodal case (2) . . . . . . . . . . . . . . . . . . . . 43 Chapter 5 Proof by induction . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.1 Energy concentration at Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Energy concentration at N . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.3 Completion of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 v LIST OF FIGURES Figure 4.1: Two families for Smooth case . . . . . . . . . . . . . . . . . . . . . . . . 39 Figure 4.2: Two families for Nodal case . . . . . . . . . . . . . . . . . . . . . . . . . 39 Figure 4.3: Coordinate expression for Smooth case . . . . . . . . . . . . . . . . . . . 44 Figure 4.4: Coordinate expression for Nodal case . . . . . . . . . . . . . . . . . . . . 47 vi Chapter 1 Introduction Harmonic map is a classical object in both geometry and PDEs, as this is a generalization of solution of Laplace equation and a minimizer of the Dirichlet energy functional. However, its compactness issue remained open until Uhlenbeck [SU] showed famous compactness theorem. Here we have to exclude finite set of points, where energy concentrates, called bubble points. Later, Parker [P] showed bubble tree extension to describe how a given sequence of harmonic maps converges over bubbles. (This idea originated from Parker and Wolfson [PW] for J- holomorphic maps.) In this chapter we briefly explain basic properties of harmonic maps, Uhlenbeck Compactness Theorem and Bubble Tree Convergence Theorem. 1.1 Background on Harmonic Maps Let (Σ, g) and (X, h) be compact Riemannian manifolds with Riemannian metrics g and h. Also assume dim(Σ) = 2. Hence Σ becomes Riemannian surface with genus g. We use the same letter g to denote Riemannian metric of Σ and genus of Σ if there is no confusion. A map f : (Σ, g) → (X, h) is harmonic if it is a critical point of the energy functional Z 1 E(f ) = L(f ) = |df |2 dvolg . (1.1) 2 Σ 1 We consider family of harmonic maps fk : (Σ, g) → (X, h) with bounded energy, E(fk ) ≤ E0 < +∞. (1.2) Using fk and g, we can define corresponding energy density measures e(fk ) on Σ by 1 e(fk ) = |dfk |2 dvolg . (1.3) 2 The harmonic map equation can be written in both intrinsic way or extrinsic way. Let {xi } be coordinate functions on Σ and {yα } be coordinate functions on X. Then the Euler- Lagrange equation for energy functional (1.1) in the above coordinate system is   τ (f )α = g ij fijα − Σ Γl f α + X Γα f β f γ = 0 (1.4) ij l βγ i j where Σ Γ is Christoffel symbol in Σ, X Γ is Christoffel symbol in X. If we embed (X, h) ,→ RN isometrically, then (1.4) can be written as τ (f ) = ∆g f + Ag (df, df ) = 0 (1.5) where ∆g is Laplace-Beltrami operator with respect to g and Ag is second fundamental form of the embedding. Now we summarize basic properties of harmonic maps. Here we follow [SU] and [P]. For more results, see [EL] or [EL2]. Suppose f : (Σ, g) → (X, h) be harmonic. Then we have following lemmas. Lemma 1.1.1. [SU, Lemma 1.3] (Conformal Invariance) The energy E(f ) in (1.1) is con- 2 formally invariant. Therefore the equations (1.4) and (1.5) are also conformally invariant. Lemma 1.1.2. [SU, Main Estimate 3.2] (ε-regularity) There is a constant ε0 > 0 depending only on second fundamental form of the embedding X ,→ RN such that if f is a harmonic map on a disk D and if ED (f ) = 21 D |df |2 dvolg < ε0 , then for any D0 ⊂⊂ D, R kdf kW 1,p (D0 ) ≤ Ckdf kL2 (D) , (1.6) where 1 < p < ∞ and C is a constant which only depends on p, D0 and the geometry of X. Lemma 1.1.3. [SU, Theorem 3.3] (Energy-gap) There is a constant ε00 > 0 depending only on (X, h) such that if f is a smooth harmonic map on a compact domain Σ satisfying E(f ) = 21 Σ |df |2 dvolg < ε00 , then f is constant. R Lemma 1.1.4. [SU, Theorem 3.6] (Removable Singularity) If f : D \ {0} → X is a C 1 harmonic map with E(f ) < ε0 on a punctured disk D \ {0}, then f can be extended to D in C 1. 1.2 Uhlenbeck Compactness Theorem and Bubble-tree extension Now consider a family of harmonic maps. First we have a convergence result. Lemma 1.2.1. [SU, Lemma 4.2] (C 1 -convergence) There is a constant ε0 > 0 such that if {fk } is a family of harmonic maps on D and satisfying ED (fk ) < ε0 for all k, then there is a subsequence fk that converges to f in C 1 . 3 Using Lemma 1.2.1, the Removable Singularity Lemma 1.1.4, and the Energy-gap Lemma 1.1.3, Uhlenbeck [SU] showed the following compactness theorem. Theorem 1.2.2. (Uhlenbeck Compactness Theorem, [SU, Theorem 4.4] or [P, Lemma 1.2]) Suppose {fk } be a sequence of harmonic maps with uniformly bounded energy. Then there are at most finite number of points {p1 , . . . , pl }, subsequence of {fk } and limit map f∞ : (Σ, g) → (X, h) such that fk → f∞ (1.7) in C 1 for any compact set away from {p1 , . . . , pl }, and Xl e(fk ) → e(f∞ ) + mi δpi (1.8) i=1 as measures where mi ≥ ε00 . In particular, Theorem 1.2.2 means that, after passing to a subsequence, we have Xl lim E(fk ) = E(f∞ ) + mi . (1.9) k→∞ i=1 The points {p1 , . . . , pl } are called bubble points. Pl After Uhlenbeck, people tried to capture the energy loss described by i=1 mi above. The problem is that a bubble may occur over the existing bubble. Parker [P] used iterated renormalizations to construct a so-called “bubble tree” to capture the energy completely. Theorem 1.2.3. (Bubble Tree Convergence, [P, Theorem 2.2]). Let {fn } : Σ → X be a sequence of harmonic maps from a fixed Riemann surface (Σ, g) to a compact Riemannian manifold (X, h) with E(fn ) ≤ E0 . Then there is a subsequence {fn } and a bubble tower 4 S domain T = Σ ∪ SI so that the renormalized maps {fn,I } : T → X (1.10) converges in W 1,2 ∩ C 0 to a smooth harmonic bubble tree map {fI } : T → X. Moreover, P 1. (No energy loss) E(fn ) converges to E(fI ), and 2. (Zero distance bubbling) At each bubble point xJ (at an level in the tree), the images of the base map fI and the bubble map fJ meet at fI (xJ ) = fJ (p− ). Consequently the image of the limit {fI } : T → X is connected, and the images of the original maps fn : Σ → X converge pointwise to this image {fI }. This method is limited to a fixed domain and can not extend to the case where the domain changes. This paper aims to generalize the result to changing domain, and provide a better strategy to capture bubble energy. This paper consists of five chapters started by introduction chapter. The second chapter deals with our main question, that is, how to generalize Bubble Tree Convergence Theorem to changing domain. To achieve this, we look over Deligne-Mumford moduli space and develop necessary convergence terminology. Then we state our main theorem and corollaries. In the third chapter, we focus on neck analysis. The main issue in this chapter is the case that energy concentrates over the neck. It is a known issue and already dealt in many other papers. We follow some papers and discover that regular node does not have such energy concentration. In fact, around the regular node, the image of fk over the neck shrinks to a point. We also cover the case of non-regular node, in which we do not have such strong convergence result. 5 Fourth chapter explains general procedure of building bigger family by putting appro- priate marked points. Additional marked points correspond to additional marked sections in the family which is bigger than before. Moreover, if two marked sections intersect in old family, we can blow-up to get bigger family together with forgetful map. This forgetful map is important in this paper because it provides coordinate transition between old and new families. The last chapter contains proof of the main theorem. First we show lemmas indicating where we put marked points in terms of energy density measure. These lemmas also tell us that on the exceptional divisor (which is a component that collapses under forgetful map), energy identity holds. Then we define residual energy, which is roughly the energy loss minus number of bubble points multiplied by quantization of minimum energy. Using this energy identity and residual energy, we can prove the main theorem by induction. In the appendix we prove Marking Points Lemmas which is technical. 6 Chapter 2 Convergence in changing domain In this chapter we define a notion of convergence for a sequence of maps fk : Ck → X , where X is a fixed Riemannian manifold. We assume that Ck are stable marked complex curves. 2.1 Families of Curves Definition 2.1.1. A smooth n-marked curve is a complex curve C (i.e. a closed 1-dimensional complex manifold) together with a sequence {x1 , . . . , xn } of distinct points of C. The points xi are called marked points. An n-marked nodal curve is a quotient C = C̃/π where C̃ is n + 2k marked curve and π is a total pairing of {xn+1 , . . . , xn+2k }. The paired points xn+i are called nodal points, or nodes. We say C̃ be a nomalization of C, and a component of C̃ is called an irreducible component of C. Each smooth connected curve has a genus g. For a connected nodal curve, we can smooth the curve and define (arithmetic) genus as that of smoothing. Definition 2.1.2. A (g, n) curve is a connected n-marked nodal curve of genus g. Definition 2.1.3. A (g, n) curve is said to be stable if 2g − 2 + n > 0. 7 As in algebraic geometry, it is useful and important to consider not just single curves, but instead families of curves. There are standard definitions of such families used by algebraic geometers. We will use an equivalent definition that is more in the spirit of differential geometry. Definition 2.1.4. ([RS] 4.2,4.7) An n-marked nodal family is a surjective proper holo- morphic map π : C → B between connected complex manifolds with disjoint submanifolds, together with N , S1 , . . . , Sn such that 1. dimC (C) = dimC (B) + 1, 2. π|Si maps Si diffeomorphically onto B, 3. N = {p ∈ C : dπ(p) not surjective}, and 4. Each critical point p ∈ N has local holomorphic coordinate chart (x, y, z2 , . . . , zn ), called a nodal chart, such that π(x, y, z2 , . . . , zn ) = (xy, z2 , . . . , zn ) (2.1) is a local holomorphic coordinate chart in a neighborhood of π(p). We call N the nodal set, and Si the marked sections. By the holomorphic Implicit Function Theorem, p ∈ C \ N has local holomorphic co- ordinate chart (x, z1 , z2 , . . . , zn ), called a regular chart, such that π(x, z1 , z2 , . . . , zn ) = (z1 , z2 , . . . , zn ) is a local holomorphic coordinate chart in a neighborhood of π(p). Note that N intersects each fiber Cb := π −1 (b) in a finite set. For each regular value b ∈ B of π the fiber Cb is a compact Riemann surface. For each critical value b ∈ B of π the fiber Cb is a nodal curve. 8 Definition 2.1.5. ([ACG] 11.2.1) Let C be a (marked) complex curve (or a nodal curve). A deformation of C is a nodal family π : C → (B, b0 ) plus a given isomorphism between C and the central fiber π −1 (b0 ) that maps marked points to marked sections. We understand the concept of deformation in terms of a germ. Hence a restriction of a deformation containing the central fiber is regarded as equivalent as the original one. Definition 2.1.6. ([RS] 4.5) Let (π : C → B, S∗ ) and (π 0 : C 0 → B 0 , S∗0 ) be nodal families. A morphism from π 0 to π is a commutative diagram Ψ C0 C π0 π (2.2) ψ B0 B where Ψ and ψ are holomorphic, the restriction Ψb0 is an isomorphism between fibers, and Ψ(S∗0 ) ⊂ S∗ . Definition 2.1.7. ([ACG] 11.4.K) A deformation π : C → (B, b0 ) of (C; p1 , . . . , pn ) is said to be a universal deformation (or Kuranishi family) for (C; p1 , . . . , pn ) if it satisfies the following condition: For any deformation π 0 : C 0 → (B 0 , b00 ) of (C; p1 , . . . , pn ), there exists a unique morphism Ψ C0 C π0 π (2.3) ψ (B 0 , b00 ) (B, b0 ) −1 such that ψ(b0 ) = b, Ψ(S 0 ) ⊂ S and ψ|C 0 : Cb0 → Cb0 = ϕ ◦ ϕ0 where S, S 0 are marked b0 0 sections of π, π 0 respectively and ϕ : C → Cb0 , ϕ0 : C → Cb0 are given isomorphisms. 0 9 Theorem 2.1.8. ([ACG] 11.4.3, 11.4.8) A Kuranishi family for (C; p1 , . . . , pn ) exists if and only if (C; p1 , . . . , pn ) is stable. The base of the Kuranishi family of a stable n-pointed curve (C; p1 , . . . , pn ) of genus g is smooth of dimension 3g − 3 + n. Corollary 2.1.9. ([ACG] 11.4.10) Let π : C → (B, b0 ) be a Kuranishi family for (C; p1 , . . . , pn ). Denote by G the automorphism group of (C; p1 , . . . , pn ). Then there are arbitrarily small neighborhoods V of b0 such that the action of G on (C; p1 , . . . , pn ) extends to compatible actions on V and on CV taking each distinguished section to itself. 2.2 Deligne-Mumford moduli space Now we describe the Deligne-Mumford moduli space. The Deligne-Mumford moduli space of stable n-marked nodal curves of genus g is defined by  Mg,n := isomorphism classes [C] of (g, n) curves . (2.4) The structure of (2.4) is well-known; see, for example, [ACG]. Here we point out prop- erties of (2.4) that will be used in this paper. • Let Mg,n be a moduli space of stable n-marked smooth curves of genus g. Then Mg,n is its compactification. • Mg,n is a complex projective variety, and has orbifold structure. • There is a projection map, called forgetful map, given by Φ : Mg,n+1 → Mg,n (2.5) 10 which forgets the last marked point and collapses unstable component to a point. There are several different ways to construct Mg,n . A key part of the construction is the fact that a neighborhood of [C0 ] in Mg,n can be constructed as a quotient of Kuranishi family of C0 by the finite group Aut(C0 ) of automorphisms of central fiber C0 . Now suppose {Ck } be a sequence of (g, n) curves and [Ck ] → [C0 ] in Mg,n . Remark 2.2.1. Even if Aut(Ck ) = 1 for all k, Aut(C0 ) may not be trivial. If Aut(C0 ) is not trivial, then the Deligne-Mumford space has an orbifold point at [C0 ]. Instead, we use Kuranishi family π : C → (B, 0) of one of representatives C0 of [C0 ] without quotient. Then the total space is a smooth manifold, but Aut(C0 ) acts as transformation of fibers as in Corollary 2.1.9, which destroys uniqueness of embedding of Ck into family. Definition 2.2.2. For sequence of (g, n) curves Ck , we say Ck → C0 in a family C if π : C → (B, 0) is a deformation of C0 with isomorphism ϕ : C0 → π −1 (0) and isomorphisms ϕk : Ck → π −1 (bk ) with bk → 0. Since ϕ and ϕk are isomorphisms, we identify Ck with π −1 (bk ) and C0 with π −1 (0). Note that Ck → C0 in a family C implies [Ck ] → [C0 ] in Mg,n . Remark 2.2.3. There is no uniqueness statement for the choice of the sequence bk and the isomorphisms ϕ and ϕk . However, all of the convergence statements below hold for any choice. 2.3 Sequences of Maps In a regular chart, we can give a local coordinate of Ck by (x, bk ) and C0 by (x, 0). Consider a sequence of smooth maps fk : Ck → X where Ck are (g, n) curves. Here smoothness means 11 that it is smooth on each irreducible component. Also, there is an energy Z 1 E(fk ) = |dfk |2 dvol (2.6) 2 C k with additional assumption that E(fk ) ≤ E0 < +∞ for all k. Definition 2.3.1. Let fk : Ck → X be a sequence of C l (or W l,p ) maps. We say fk converges to f0 : C0 → X off the set S ⊂ C0 in C l (or W l,p ) if 1. Ck → C0 , 2. S includes all nodal points, and 3. in every regular chart away from S, the projected map f˜k (x) := fk (x, bk ) (2.7) converges to f˜0 (x) := f0 (x, 0) in C l (or W l,p ). The set S is called singular set. Definition 2.3.2. Let Ck → C0 and p be a node of C0 . We say a sequence of maps fk : Ck → X satisfies the zero neck property at p if the following holds: For any ε > 0, there is δ1 > 0 such that for any 0 < δ < δ1 and for all k sufficiently large, E(fk , Ck ∩ B(p, δ)) ≤ ε (2.8) diam(fk (Ck ∩ B(p, δ))) ≤ ε. (2.9) 12 Remark 2.3.3. Note that the zero neck property holds for J-holomorphic maps, which are harmonic maps. Definition 2.3.4. Let fk : Ck → X be a sequence of C l (or W l,p ) maps. We say fk converges to f0 : C0 → X in C l (or W l,p ) if 1. fk converges to f0 off the singular set S ⊂ C0 in C l (or W l,p ), and 2. S only consists of nodal points where fk satisfies the zero neck property. Lemma 2.3.5. (Energy Identity and Connected Image) Suppose fk : Ck → X converges to f0 : C0 → X in C 1 . Then lim E(fk ) = E(f0 ). (2.10) k→∞ Furthermore, if each Ck is connected, then the image of f0 is connected. Proof. Pick any regular point p ∈ C0 \ S. Let U ∈ C0 be a neighborhood of p and consider the regular chart π : U × B → B given by π(x, b) = b with p = (0, 0). In the regular chart, the projection map πk : U × {bk } → U given by πk (x, bk ) = x is holomorphic, so Z Z 1 2 1 E(fk , U × {bk }) = |dfk | dvol = |df˜k |2 dvol 2 U ×{b } 2 U k where f˜k (x) = fk ◦ πk (x) = fk (x, bk ). Since fk converges to f0 off the singular set S ⊂ C0 in C 1 and p ∈/ S, lim E(fk , U × {bk }) = E(f0 , U ). k→∞ Also note that since S only consists of nodes where fk satisfies the zero neck property, lim lim E(fk , Ck ∩ B(q, δ)) = 0 δ→0 k→∞ 13 for any q ∈ S. Cover C0 \ S by finitely many regular charts {Ui × B} then we have lim E(fk ) = E(f0 ). k→∞ Connectedness comes from the other condition of the zero neck property. 2.4 Main Theorem In this section we deal with our original question. Question 2.4.1. Let fk : Ck → X be a sequence of harmonic maps with uniformly bounded energy where Ck are (g, n) curves. In this setting, how do we define convergence of fk ? Is there any finite singular set S such that fk converge away from S? What do we know about S? Under which condition does fk converge, i.e., without singular set S? Our Main Theorem 2.4.6 and its corollary give complete answer of the question. It shows that, by adding more marked points appropriately, one can construct a family on curves and show that fk converges off the non-regular nodes. Moreover, with additional assumption, fk converges everywhere (i.e. with empty singular set). The proof begins by applying Uhlenbeck’s compactness theorem 1.2.2. Define the energy density of a map f to be e(f ) = 12 |df |2 dvol, regarded as a measure on domain. Also suppose that Ck → C0 for some nodal (g, n) curve C0 and denote N be a nodal set in C0 . Note that in regular chart f˜k (x) := fk (x, bk ) as in (2.7). Lemma 2.4.2. (Local convergence) There is a subsequence nk , a finite set Q = {p1 , . . . , pl } ⊂ 14 C0 \ N , and a harmonic map f0 : C0 → X such that f˜nk converges to f0 in C 1 on any com- pact set K ⊂ C0 \ (Q ∪ N ). Also, near p ∈ Q, we have e(f˜nk ) → e(f0 ) + mp δp , (2.11) where δp is a Dirac-delta measure at p and mp ≥ ε00 . Proof. Cover C0 \ N by finitely many regular chart {Ui × B}. Since the regular chart is holomorphic coordinate chart, f˜k (x) is also harmonic maps over Ui for each i. By Uhlenbeck compactness theorem 1.2.2, there is a subsequence nk,1 such that f˜nk,1 converges on U1 \ Q1 where Q1 is a finite set. Applying Theorem 1.2.2 again to find subsequence nk,2 of nk,1 such that f˜nk,2 converges on U2 \ Q2 where Q2 is a finite set. Repeat this process to obtain nk,i for Ui , and choose diagonal subsequence nk = nk,k . Then f˜nk converges on Ui \ Qi for all i, where Qi is a finite set. Note that at each point p ∈ Qi , by (1.8), the energy concentration at p is at least ε00 . Since the total energy is finite, Q = ∪Qi is at most finite. The measure convergence (2.11) comes from (1.8). The above lemma says that fnk converges to f0 off the singular set S := Q ∪ N and that energy loss may occur only at points in S. Definition 2.4.3. In Lemma 2.4.2, we say p ∈ Q a smooth bubble point and p ∈ N a nodal bubble point. Now we will put two marked points both converging to each point in Q to build bigger family of curves. Then Q becomes a nodal point in this bigger family of curves. We will describe details of the procedure of adding more marked points in Chapter 4. Also note that from the Lemma 1.1.3, energy concentration at Q is at least ε00 . This gives 15 quantization of energy level of nontrivial component, and becomes one of key ingredients to prove Main Theorem 2.4.6. Moreover, on the neighborhood of the regular node, there is another quantized energy level ε000 described as follows. First we define regular node. Let π : C → B be a deformation of (g, n) curves C0 and p be a node of C0 . Definition 2.4.4. We say p is a regular node if there exists a family of (g, l) curves π : C → B with l < n and a forgetful map Φ : C → C such that p := Φ(p) is a regular point. Lemma 2.4.5. (Energy gap in the neck) There is ε000 > 0 such that the following holds: Suppose E(fk , Ck ∩ B(p, δ0 )) ≤ ε000 for some δ0 > 0 and for all k near regular node p. Then fk satisfies the zero neck property (Definition 2.3.2). Proof of this lemma will be given in Chapter 3. Now we state our main theorems. Theorem 2.4.6. (Main Theorem) Suppose fk : Ck → X be a sequence of harmonic maps with uniformly bounded energy defined on smooth (g, n) curves. Then there is a subsequence nk and a way of marking points Pk on Ck such that cor- responding sequence fnk : Cn0 = (Cnk , Pnk ) → X converges to some f0 : C0 → X off the k singular set S in C 1 where all points in S are non-regular nodal points. Furthermore, f0 is harmonic on closure of each component of C0 \ S separately. Corollary 2.4.7. Suppose fk : Ck → X be a sequence of harmonic maps with uniformly bounded energy defined on smooth (g, n) curves. Also assume Ck → C0 in a family C and all nodes in C0 are regular. 16 Then there is a subsequence nk and a way of marking points Pk on Ck such that corre- sponding sequence fnk : Cn0 = (Cnk , Pnk ) → X converges to some f0 : C00 → X in C 1 where k Cn0 → C00 in bigger family C 0 . That means, the singular set S in Theorem 2.4.6 is empty. k Furthermore, f0 is harmonic on each irreducible component of C0 , the energy identity holds, and the image of f0 is connected. Special case of Corollary 2.4.7 is when domain is fixed. This is actually classical bubble- tree convergence theorem as Parker showed [P]. Corollary 2.4.8. Let Σ be a smooth Riemann surface with genus g and suppose fk : Σ → X be a sequence of harmonic maps with uniformly bounded energy. Then there is a subsequence nk , a way of marking points Pk on Σ, a marked nodal curve C0 , and a limit map f0 : C0 → X such that corresponding sequence fnk : Cnk = (Σ, Pnk ) → X converges to f0 in C 1 . Furthermore, f0 is harmonic on each irreducible component of C0 , the energy identity holds, and its image is connected. Proofs of Main Theorem and corollaries will be given in Section 5.3. 17 Chapter 3 Neck analysis In this chapter we will deal with sequence of harmonic maps over the neck region. The goal of this chapter is to show Lemma 2.4.5. For convenience, we recall the statement: Lemma 2.4.5. (Energy gap in the neck) There is ε000 > 0 such that the following holds: Suppose E(fk , Ck ∩ B(p, δ0 )) ≤ ε000 for some δ0 > 0 and for all k near regular node p. Then fk satisfies the zero neck property (Definition 2.3.2). Now we will set up the terminology. Consider a sequence of harmonic maps fk : Ck → X defined on family of stable curves Ck with Ck → C0 where C0 is a limit curve with a node p. For any δ > 0, using nodal chart, we can denote Ck ∩ B(p, δ) = {x, y ∈ C2 : xy = tk , |x|, |y| ≤ δ} and tk → 0 as k → ∞. Consider polar coordinate x = (r, θ), and take log by √ (r, θ) → (t, θ) where t = ln(r/ tk ). This gives a cylindrical coordinate over the neck which is conformal to the original nodal chart, given by φ : [−Tkδ , Tkδ ] × S 1 → Ck ∩ B(p, δ) (3.1) √ √ √ where φ(t, θ) = (x, y) = ( tk et+iθ , tk e−t−iθ ) and Tkδ = ln(δ/ tk ). Note that pullback of the metric to the cylinder is conformally equivalent to dt2 +dθ2 , so we can consider flat metric over the cylinder. For simplicity, we also use fk for fk ◦ φ. Also, if there is no confusion, we 18 sometimes omit subscription k. Throughout this chapter, we will use both nodal chart or cylindrical chart suitably. 3.1 General neck analysis In this section we will show several lemmas to show Lemma 3.1.8. This lemma controls the energy and diameter of the neck in terms of α (defined in (3.4)). For details, see [P] or [Z2]. Harmonic map equation for f : [−Tkδ , Tkδ ] × S 1 → X ,→ RN can be written by ∆f = ftt + fθθ = −A(f )(df, df ) ⊥ Tf X (3.2) where A(f ) is second fundamental form of the embedding, and the energy is given by ZZ 1 E(f ) = |ft |2 + |fθ |2 dt dθ. (3.3) 2 [−T δ ,T δ ]×S 1 k k Lemma 3.1.1. [Z2, Lemma 3.3] The quantity Z 1 α(t) := |ft |2 − |fθ |2 dθ (3.4) 2 {t}×S 1 is independent of t. 19 Proof. Since ∆f ⊥ ft , for any T1 < T2 , we have ZZ ZZ 0= h∆f, ft idθ dt = hftt + fθθ , ft idθ dt [T1 ,T2 ]×S 1 [T1 ,T2 ]×S 1 ZZ   ZZ 1 2 = ∂t |ft | dθ dt − hfθ , ftθ idθ dt [T1 ,T2 ]×S 1 2 [T1 ,T2 ]×S 1 ZZ   1 2 1 2 = ∂t |ft | − |fθ | dθ dt [T1 ,T2 ]×S 1 2 2 Z T 2 = ∂t αdt = α(T2 ) − α(T1 ). T1 Therefore α(T2 ) = α(T1 ) and α is independent on t. Remark 3.1.2. In [Z2], the author uses different definition of α which is a complex number. In fact, our α is the real part of α in [Z2]. We can recover equation for the imaginary part by using fθ from the above proof instead of ft , but we do not use it here. From Lemma 3.1.1, the energy can be written as Z Tδ ZZ Z Tδ k k E(f ) = αdt + |fθ |2 dt dθ = 2Tkδ α + Θdt (3.5) −T δ [−T δ ,T δ ]×S 1 −T δ k k k k where Z Θ(t) := |fθ |2 dθ. (3.6) {t}×S 1 Lemma 3.1.3. ([Z2, Lemma 3.1] or [P, Lemma 3.2]) There exists ε000 > 0 such that if f : [−Tkδ , Tkδ ] × S 1 → X is a harmonic map and E(f ) ≤ ε000 , then Θ00 ≥ Θ > 0. (3.7) Proof. First, by applying ε-regularity to B2 (t, θ), we have sup |df |2 ≤ CE(f ) ≤ Cε000 . B1 (t,θ) 20 q Since the center point (t, θ) is arbitrary, we actually have sup norm bound sup|df | ≤ Cε000 . Now, d2 Z Z Z Θ00 = 2 2 |f | dθ = 2 2 |fθt | dθ + 2 hfθ , fttθ idθ dt {t}×S 1 θ {t}×S 1 {t}×S 1 Z Z =2 2 |fθt | dθ − 2 hfθθ , ftt idθ {t}×S 1 {t}×S 1 Z Z Z =2 2 |fθt | dθ + 2 2 |fθθ | dθ − 2 hfθθ , A(f )(df, df )idθ. {t}×S 1 {t}×S 1 {t}×S 1 The last term, denoted by III, becomes Z Z III = −2 hfθθ , A(f )(df, df )idθ = 2 hfθ , (A(f )(df, df ))θ idθ {t}×S 1 {t}×S 1 Z Z =2 hfθ , ∇A(df, df ) · fθ idθ + 2 2hfθ , A(f )(fθθ , fθ ) + A(f )(fθt , ft )idθ. {t}×S 1 {t}×S 1 So, Z |III| ≤ 2k∇Ak∞ sup|df |2 |fθ |2 dθ {t}×S 1 Z Z + 4kAk∞ |fθθ ||fθ |2 dθ + 4kAk ∞ |fθt ||fθ ||ft |dθ {t}×S 1 {t}×S 1 q Z q Z q Z ≤ C(ε000 + ε0 ) 00 |fθ |2 dθ + C ε000 |fθθ |2 dθ + C ε000 |fθt |2 dθ. {t}×S 1 {t}×S 1 {t}×S 1 Here C is a constant depending only on the geometry of X. Therefore, if we choose ε000 small enough, we have Z Z Z 3 1 Θ00 ≥ 2 |f | dθ − 2 |f | dθ ≥ |fθ |2 dθ = Θ 2 {t}×S 1 θθ 2 {t}×S 1 θ {t}×S 1 21 where the last inequality comes from Poincaré inequality on S 1 . Lemma 3.1.4. [Z2, Lemma 3.1] Suppose f : [−Tkδ , Tkδ ] × S 1 → X is a harmonic map and E(f ) ≤ ε000 . Then for any −Tkδ ≤ T1 < T2 ≤ Tkδ , we have Z T 2 Θdt ≤ 2(Θ(T1 ) + Θ(T2 )), (3.8) T1 Z T2 √ p p Θdt ≤ 4( Θ(T1 ) + Θ(T2 )). (3.9) T1 Proof. Let τi = Θ(Ti ) for i = 1, 2. Since Θ00 ≥ Θ, by maximum principle, we have Θ ≤ Aet + Be−t where eT2 τ2 − eT1 τ1 e−T2 τ2 − e−T1 τ1 A= , B= . e2T2 − e2T1 e−2T2 − e−2T1 Also, Θ1/2 ≤ |A|1/2 et/2 + |B|1/2 e−1/2 . Therefore, for λ = 1/2 or 1, − eλT1 e−λT1 − e−λT2 Z T 2 eλT2 Θλ dt ≤ |A|λ + |B| λ T1 λ λ eλT2 − eλT1 −λT1 − e−λT2 ≤ |eT2 τ2 − eT1 τ1 |λ + |e −T2 τ − e−T1 τ |λ e 2 1 λ(e2T2 − e2T1 )λ λ(e−2T1 − e−2T2 )λ eλT2 − eλT1 e−λT1 − e−λT2 ≤ (eλT2 τ2λ + eλT1 τ1λ ) + (e−λT2 τ2λ + e−λT1 τ1λ ) λ(e2λT2 − e2λT1 ) λ(e−2λT1 − e−2λT2 ) 2 λ ≤ (τ + τ2λ ) λ 1 which proves the lemma. Lemma 3.1.5. Suppose f : [−Tkδ , Tkδ ] × S 1 → X is a harmonic map and E(f ) ≤ ε000 . For 22 any ε > 0, there is δ2 > 0 such that for any 0 < δ < δ2 and for all k sufficiently large, Z Tδ Z Tδ √ k k Θdt ≤ ε, Θdt ≤ ε. (3.10) −T δ −T δ k k Proof. First, choose δ2 > 0 such that limit map f0 has energy less than ε/2 over the region C0 ∩ (Bδ \ Bδ/2 ) = {(0, y) ∈ C2 : δ/2 ≤ |y| ≤ δ} ∪ {(x, 0) ∈ C2 : δ/2 ≤ |x| ≤ δ} for δ < δ1 . By local convergence lemma 2.4.2, for δ < δ2 and for all k sufficiently large, fk has energy less than ε over the region Ck ∩ (Bδ \ Bδ/2 ) = {(x, y) ∈ C2 : xy = tk , δ/2 ≤ |x|, |y| ≤ δ}. Using cylindrical domain, this energy bound can be written by Z −T δ +ln 2 Z Z Tδ Z k k Ek (Bδ \Bδ/2 ) := |ft |2 +|fθ |2 dθ dt+ |ft |2 +|fθ |2 dθ dt ≤ ε. (3.11) −T δ S1 T δ −ln 2 S 1 k k Now fix δ, k. Choose a ∈ (−Tkδ , −Tkδ + ln 2), b ∈ (Tkδ − ln 2, Tkδ ) such that Z −T δ +ln 2 Z Tδ 1 k 1 k Θ(a) = Θdt, Θ(b) = Θdt. ln 2 −T δ ln 2 T δ −ln 2 k k Then, by Lemma 3.1.4, Z Tδ Z a Z Tδ Z b k k Θdt = Θdt + Θdt + Θdt −T δ −T δ b a k k Z −T δ +ln 2 Z Tδ k k ≤ Θdt + Θdt + 2(Θ(a) + Θ(b)) −T δ T δ −ln 2 k  k  Z −T δ +ln 2 Z Tδ 2  k k = (1 + ) Θdt + Θdt ln 2 −T δ T δ −ln 2 k k 2 2 ≤ (1 + )Ek (Bδ \ Bδ/2 ) ≤ (1 + )ε. ln 2 ln 2 23 Second inequality can be obtained in the similar manner. Since δ and k is arbitrary provided δ < δ1 and k sufficiently large, these estimates are valid for any 0 < δ < δ1 and for all k sufficiently large. Lemma 3.1.6. Suppose f : [−Tkδ , Tkδ ] × S 1 → X is a harmonic map and E(f ) ≤ ε000 . For any ε > 0, there is δ3 > 0 such that for any 0 < δ < δ3 and for all k sufficiently large, Z |fθ |dθ ≤ ε. (3.12) {t}×S 1 Proof. Consider nodal chart of the neck Ck ∩ B(p, δ). Choose δ3 > 0 such that the length of f0 (|x| = δ) and f0 (|y| = δ) are less than ε/2 for δ < δ3 . Then for all k sufficiently large, the length of fk (|x| = δ) and fk (|y| = δ) are less than ε. Using cylindrical coordinate, this means Z |fθ |dθ ≤ ε. {±T δ }×S 1 k Now, by ε-regularity, !1/2 Z √ p √ q √ Z |fθ |dθ ≤ 2π Θ(t) ≤ 2π Θ(±Tkδ ) = 2π |fθ |2 dθ {t}×S 1 δ {±T }×S 1 k !1/2 √ p Z ≤ 2π sup|df | |fθ |dθ {±T δ }×S 1 k q √ ≤C ε000 ε. 24 To control diameter, we first define average length of the neck as follows: ZZ 1 L := |ft |dtdθ. (3.13) 2π [−T δ ,T δ ]×S 1 k k In general, smallness of diameter of image is irrelevant with smallness of L. (One can construct a map defined on cylinder with image of small diameter and large L, and a map with small L and image having large diameter.) However, we show that for harmonic maps, smallness of L implies smallness of diameter. Lemma 3.1.7. Suppose f : [−Tkδ , Tkδ ] × S 1 → X is a harmonic map and E(f ) ≤ ε000 . For any ε > 0, there is δ2 > 0 such that for any 0 < δ < δ3 and for all k sufficiently large, diamf = max (|f (x) − f (y)|) ≤ 2ε + L. (3.14) x,y∈[−T δ ,T δ ]×S 1 k k Proof. First, by mean value theorem, choose θ0 such that L = Lθ0 . Now for given ε > 0, choose δ3 as in Lemma 3.1.6. Let (t1 , θ1 ) and (t2 , θ2 ) be such that max (|f (x) − f (y)|) = |f (t1 , θ1 ) − f (t2 , θ2 )|. x,y∈[−T δ ,T δ ]×S 1 k k Then, diamf = |f (t1 , θ1 ) − f (t2 , θ2 )| ≤ |f (t1 , θ1 ) − f (t1 , θ0 )| + |f (t1 , θ0 ) − f (t2 , θ0 )| + |f (t2 , θ0 ) − f (t2 , θ2 )| Z Z ≤ |fθ |dθ + Lθ0 + |fθ |dθ {t1 }×S 1 {t2 }×S 1 ≤ 2ε + L. 25 Lemma 3.1.8. Suppose fk : [−Tkδ , Tkδ ] × S 1 → X is a sequence of harmonic maps and E(fk ) ≤ ε000 for all k. For any ε > 0, there is δ1 > 0 such that for any 0 < δ < δ1 and for all k sufficiently large, E ≤ 2Tkδ α + ε, √ diamf ≤ C αTkδ + ε. Proof. Let δ1 = min{δ2 , δ3 }. The first equation comes from Equations (3.5) and Lemma 3.1.5. Note that ZZ Z T δ √ Z 1/2 1 1 k 2 L= |ft |dt ≤ 2π |ft | dθ dt 2π [−T δ ,T δ ]×S 1 2π −T δ S1 k k k Z T δ √ Z 1/2 Z 1/2 1 k 2 2 2 ≤ 2π |ft | − |fθ | dθ + |fθ | dθ dt 2π −T δ S1 S1 k Z Tδ √ 2 √ δ 1 k √ =√ αTk + Θdt ≤ C αTkδ + ε 2π 2π −T δ k by Lemma 3.1.5. Now applying Lemma 3.1.7, we get the second inequality. 3.2 Regular node In general, α is not small enough to obtain the zero neck property. However, if the node is regular, α vanishes. In this section we will show that if the node is regular, we have the zero neck property. Lemma 3.2.1. Suppose fk : Ck → X is a sequence of harmonic maps, Ck → C0 and p ∈ C0 26 is a regular node. Then in the cylindrical coordinate, Z 2π Z 2π |ft |2 dθ = |fθ |2 dθ (3.15) 0 0 which means that α = 0. Proof. Let π : C → B be a deformation of C0 . Let Φ : C → C be a forgetful map such that p = Φ(p) is a regular point. Note that the restriction of Φ to Ck is a biholomorphism to Ck := Φ(Ck ) except forgetting n − l marked points. The nodal coordinate of Ck near p is given by (x, y) such that xy = tk for some tk . Choose a regular chart z of Ck around p. Without loss of generality, Φ can be written as Φ(x, y) = z where z = x over the nodal chart around p. Denote Fk := fk ◦ Φ−1 : Ck → X, which is also harmonic. Let D be a disk centered at p in the regular chart of Ck . By Pohozaev identity, Z 2π Z 2π 2 2 Fφ dφ = r2 Fr dφ (3.16) 0 0 where (r, φ) is the polar coordinate on D. For the proof of this identity, see [SU, Lemma 3.5] or [LW, Lemma 6.1.5]. Recall the cylindrical coordinate φ : [−Tkδ , Tkδ ] × S 1 → Ck given by (3.1). Then Φ ◦ φ : √ [−Tkδ , Tkδ ] × S 1 → Ck gives a coordinate change given by Φ ◦ φ(t, θ) = z = tk et+iθ , so tk et Fr = rFr , p (F ◦ Φ ◦ φ)t = (F ◦ Φ ◦ φ)θ = Fφ . (3.17) Then (3.16) becomes Z 2π Z 2π 2 2 (F ◦ Φ ◦ φ)θ dθ = (F ◦ Φ ◦ φ)t dθ. (3.18) 0 0 27 Since F ◦ Φ ◦ φ = f ◦ φ, this proves the lemma. Remark 3.2.2. In the above proof, the fact that we know coordinate expression of Φ precisely is crucial. In other words, only knowing holomorphicity of Φ is not enough to get α = 0 since α is not conformally invariant. Proof. (Proof of Lemma 2.4.5) From Lemma 3.1.8, for any ε > 0, there is δ1 > 0 such that for any 0 < δ < δ1 and for all k sufficiently large, E ≤ 2Tkδ α + ε, √ diamf ≤ C αTkδ + ε. Now by Lemma 3.2.1, α = 0 and the zero neck property is satisfied. 3.3 non-regular node I - counterexample In this section we will recall Parker’s torus example. For details, see [P], Section 5. This example shows that for general sequence of harmonic maps, • the zero neck property may fail, and • there exist non-regular nodes where C0 convergence fails. In Section 5 of [P], Parker constructed a sequence of harmonic maps fn : (T 2 , hn ) → X = S 2 × S 1 with E(fn ) ≤ E0 such that the energy loss over the neck occurs in the limit. For k > 0, let cn(t, k) and sn(t, k) denote Jacobi elliptic functions, and K = K(k) is the 28 complete elliptic integral Z 1 dx K= p . (3.19) 0 (1 − x2 )(1 − k 2 x2 ) The construction starts from looking for harmonic maps from R × S 1 into S 2 × R ⊂ R3 × R. Fixing the metric dt2 + dθ2 on R × S 1 , we have a 2-parameter family of harmonic maps fk,a (t, θ) = (cn(t/k, k) cos θ, cn(t/k, k) sin θ, sn(t/k, k), at). (3.20) Now as k approaches 1 the period 4K becomes arbitrarily large. Thus given a positive integer p and a real number α > 0 we can choose an increasing sequence kn → 1 with 4Kn kn = n2 /α and an = α/np where Kn = K(kn ). After projection S 2 × R → S 2 × R/Z, we have a sequence of harmonic maps fn : R × S 1 → S 2 × S 1 that are periodic in t with period 4Kn kn p. Example 3.3.1. [P, Lemma 5.2] Let Tn be the torus S 1 × S 1 with the metric hn = dt2 + (4Kn kn p)−2 dθ2 = dt2 +(α/n2 p)2 dθ2 , and let X = S 2 ×S 1 be the product of the unit 2-sphere and the circle of length 1. Then the functions fn (t, θ) = (cn(4Kn pt, kn ) cos θ, cn(4Kn pt, kn ) sin θ, sn(4Kn pt, kn ), nt) (3.21) give harmonic maps fn : Tn → X with E(fn ) ≤ E0 . Proof. See [P], Section 5. Now consider the sequence of harmonic maps fn with n = 2pm + 1 where fn is defined 29 in Example 3.3.1. The energy density of fn is 1h i e(fn )(t) = |dft |2 + |dfθ |2 (4Kn kn p)2 dt dθ 2 1 − kn2   α 2 = + 2Kn kn p + 2 cn (4Kn pt, kn ) dt dθ. 2p kn2 Let e(fn )(t) = Fn (t)dt dθ, then |Fn | ≥ α/2p and 2l − 1 α lim Fn ( ) = , (3.22) n→∞ 4p 2p l lim Fn ( ) =∞ (3.23) n→∞ 2p for l = 0, 1, . . . , 2p − 1. Hence energy density has 2p bumps at 2p l and we can divide the domain into two types of regions, one is bead domains Bl (t near 2p l ) and the other is string domains Sl (rest of the domain including t = 2l−1 4p ). On the bead domain, by composing conformal change of domain, we have a sequence of renormalized bead maps which converge to f∞ l = ((−1)l Id., l/2p) : S 2 → S 2 × S 1 uniformly on compact sets in S 2 − {(0, 0, ±1)}. (See [P, Lemma 5.3].) On the string domain, the image is a thin tube wrapping the S 1 factor of X = S 2 × S 1 more than m times. Moreover, as n → ∞ this tube has circumference converges to zero and energy Z πα e(fn ) → . (3.24) Sl 2p2 (See [P, Lemma 5.4]). The total energy of fn now converges to πα πα E(fn ) → 2p(4π) + 2p( 2 ) = 8pπ + = E0 < +∞. (3.25) 2p p 30 Example 3.3.1 shows that the energy might not converge to zero when the metric degen- erates. Moreover, we can assign any positive amount of energy over the neck. Remark 3.3.2. By looking at the (average) length of the neck, one can analyzes the situation with more details. For example, while Example 3.3.1 gives (I) nonzero energy and infinite length of the neck, its modifications can generate (II) zero energy and infinite length of the neck, (III) zero energy and finite length of the neck, and (IV) zero energy and zero length of the neck. (See [Z2] for details.) Remark 3.3.3. On the other hand, the image of fn wraps S 1 factor of X = S 2 ×S 1 more and more as n → ∞. This means that in the homotopy class of the map space M aps(T 2 , X) = {f : T 2 → X}, [fn ] are all discrete and hence can not converge. 3.4 non-regular node II - monotonicity condition In this section, we investigate when non-regular node satisfies the zero neck property. Mo- tivated by [PW], we consider monotonicity condition. Definition 3.4.1. We say f : Ω → X satisfies monotonicity property if there are constants C > 0 and r0 > 0 only depending on geometry of X such that for any p ∈ f (Ω) and for all 0 < r < r0 , we have Area(f (Ω) ∩ B(p, r)) ≥ Cr2 . (3.26) We also say a family of maps {f : Ω → X} satisfies monotonicity property if all f in the family satisfy monotonicity property with the same constants C and r0 . We will prove the following lemma: 31 Lemma 3.4.2. Suppose fk : [−Tkδ , Tkδ ] × S 1 → X is a harmonic map and E(fk ) ≤ ε000 for all k. Also suppose fk satisfies monotonicity property for all δ and k. Then fk satisfies the zero neck property. Lemma 3.4.3. Suppose f : [−Tkδ , Tkδ ] × S 1 → X is a harmonic map and E(f ) ≤ ε000 . For any ε > 0, there is δ1 > 0 such that for any 0 < δ < δ1 and for all k sufficiently large, q Area(f ) ≤ ε000 ε. (3.27) Proof. By Lemma 3.1.5, ZZ q ZZ Area(f ) = 2 2 2 |ft | |fθ | − hft , fθ i dt dθ ≤ |ft ||fθ |dt dθ [−T δ ,T δ ]×S 1 [−T δ ,T δ ]×S 1 k k k k ZZ !1/2 ZZ !1/2 ≤ |ft |2 dt dθ |fθ |2 dt dθ [−T δ ,T δ ]×S 1 [−T δ ,T δ ]×S 1 k k k k  1/2 √ Z Tδ q k ≤ E Θdt ≤ ε000 ε. −T δ k Lemma 3.4.4. Suppose fk : [−Tkδ , Tkδ ] × S 1 → X is a harmonic map and E(fk ) ≤ ε000 for all k. Also suppose fk satisfies monotonicity property for all δ and k. Then lim lim diamfk = lim lim max (|fk (x) − fk (y)|) = 0. (3.28) δ→0 k→∞ δ→0 k→∞ x,y∈[−T δ ,T δ ]×S 1 k k 32 Proof. Suppose lim lim diamfk = d > 0 for some d. Choose δ 0 > 0 such that δ→0 k→∞ Z d lim diamfk ≥ d, lim |∂θ fk |dθ ≤ k→∞ k→∞ {±T δ }×S 1 20 k for all δ < δ 0 . Hence, for all k sufficiently large, Z 9d d diamfk ≥ , |∂θ fk |dθ ≤ 10 {±T δ }×S 1 10 k for all δ < δ 0 . Let x and y such that diamfk = |fk (x) − fk (y)|. Case 1: d(fk (x), fk ({±Tkδ } × S 1 )) ≥ d5 or d(fk (y), fk ({±Tkδ } × S 1 )) ≥ d5 for infinitely many k. Then choose p = fk (x) or p = fk (y) and let r = 10 d . Then B(p, r) ∩ fk (Ck ) ⊂ fk ([−Tkδ , Tkδ ] × S 1 ). (3.29) C 2 d4 Choose ε < . Then we have corresponding δ1 , and for all δ < δ1 and for all k sufficiently 104 ε000 large, by Lemma 3.4.3, q Cr2 ≤ Area(B(p, r) ∩ fk (Ck )) ≤ Area(fk ([−Tkδ , Tkδ ] × S 1 )) ≤ ε000 ε < Cr2 (3.30) which is a contradiction. Case 2: d(fk (x), fk ({Tkδ } × S 1 )) ≤ d5 and d(fk (y), fk ({−Tkδ } × S 1 )) ≤ d5 (or exchange of x and y) for infinitely many k. 33 In this case, d(fk ({Tkδ } × S 1 ), fk ({−Tkδ } × S 1 )) ≥ |fk (x) − fk (y)| − d(fk (x), fk ({Tkδ } × S 1 )) − d(fk (y), fk ({−Tkδ } × S 1 )) − diam(fk ({Tkδ } × S 1 )) − diam(fk (−{Tkδ } × S 1 )) 3d ≥ . 10 Now choose p ∈ fk ([−Tkδ , Tkδ ] × S 1 ) such that d(p, fk ({Tkδ } × S 1 )) ≥ 3d δ 20 and d(p, fk ({−Tk } × S 1 )) ≥ 3d d 20 and let r = 10 . Then we have (3.29). The rest argument is similar with the above, and we get a contradiction. Case 3: d(fk (x), fk ({Tkδ } × S 1 )) ≤ d5 and d(fk (y), fk ({Tkδ } × S 1 )) ≤ d5 (or for −Tkδ ) for infinitely many k. This case is impossible because 9d ≤ |fk (x) − fk (y)| 10 d ≤ d(fk (x), fk ({Tkδ } × S 1 )) + d(fk (y), fk ({Tkδ } × S 1 )) + diam(fk ({Tkδ } × S 1 )) ≤ . 2 For all cases we get a contradiction. Therefore the lemma is proved. Lemma 3.4.5. [J2, Theorem 2.5.2] Let u : D → N be a harmonic map, N be a Riemannian manifold, and B(x0 , R0 ) ⊂ D. Also assume u(B(x0 , R0 )) ⊂ B(p, ρ) for some p ∈ N with ρ < min(π/2κ, ι(p)) where K is the sectional curvature of N , κ2 is the upper bound of K (K ≤ κ2 ), and ι(p) is the injectivity radius of p. Then for all R ≤ R0 , d(u(x), u(x0 )) |du(x0 )| ≤ C max (3.31) x∈B(x0 ,R) R 34 where C depends on geometry of N . Corollary 3.4.6. Let f : [−Tkδ , Tkδ ] × S 1 → X be a harmonic map. Assume that image of f is within some ball of radius ρ < min(π/2κ, ι(X)) where K is the sectional curvature of N , κ2 is the upper bound of K (K ≤ κ2 ), and ι(X) is the injectivity radius of X. Then, !2 diamf α≤C . (3.32) Tkδ diamf Proof. From Lemma 3.4.5 with some modifications, we can get |ft |t=0 ≤ C , hence Tkδ Z Z !2 diamf α= |ft |2 − |fθ |2 dθ ≤ |ft |2 dθ ≤ C . {0}×S 1 {0}×S 1 Tkδ Remark 3.4.7. The above corollary holds when image of f is in a ball of radius ρ < min(π/2κ, ι(X)). Example 3.3.1 is a counterexample, where diamf is bounded but L can be arbitrarily large. Proof. (Proof of Lemma 3.4.2) From Lemma 3.1.8, for any ε > 0, there is δ1 > 0 such that for any 0 < δ < δ1 and for all k sufficiently large, E ≤ 2Tkδ α + ε, √ diamf ≤ C αTkδ + ε. 35 Note that from Lemma 3.4.4, diamf ≤ ε. Furthermore, by Lemma 3.4.6, (diamf )2 E ≤ 2Tkδ α + ε ≤ C +ε≤ε Tkδ which proves the lemma. 36 Chapter 4 Adding marked points to build new family In this chapter we describe the general procedure of adding marking points to build bigger family. Suppose that Ck → C0 for some nodal (g, n) curve C0 and denote N be a nodal set in C0 . By adding i more marked points on Ck , this sequence now lies in Mg,n+i and hence has different limit C00 of (g, n + i) curve. So we need to describe • how the old and new limit curve are related, and • how the coordinate expression of new family relates to the expression of old family, using the forgetful map. Note that throughout this chapter, we do not need fk . 4.1 New family and forgetful map Fix p ∈ C0 . Then either 1. Smooth case : p is a smooth point, or 2. Nodal case : p ∈ N is a nodal point. 37 Then we put 1 or 2 marked points to make a new family with new limit C00 which can be seen as a blow up of C0 at p. First assume that p is not marked. Lemma 4.1.1. Fix a deformation π : C → B and suppose Ck = π −1 (bk ), C0 = π −1 (0) with bk → 0. Further suppose for each k that 1. For Smooth case (1), there are two unmarked distinct points qk , rk ∈ Ck such that p = lim qk = lim rk . (4.1) k→∞ k→∞ 2. For Nodal case (2), there is an unmarked point rk ∈ Ck such that p = lim rk . (4.2) k→∞ Denote Ck0 = (Ck , qk , rk ) for (1), or Ck0 = (Ck , rk ) for (2). Then there is a subsequence nk , a nodal (g, n + i) marked curve C00 where i = 1 for (1) and i = 2 for (2), a deformation π 0 : C 0 → B 0 of C00 , and forgetful map Φ C0 C π0 π (4.3) B0 B such that Cn0 = (π 0 )−1 (b0k ) with b0k → 0, Φ(Cn0 ) = Cnk , and Φ(C00 ) = C0 . Here C 0 comes k k with two additional sections σ and τ corresponding to qk and rk for Smooth case (1), and with one additional section τ corresponding to rk for Nodal case (2). Furthermore, the restriction of Φ on C00 is the map collapsing a rational curve E, and 38 is biholomorphic on C00 \ E. In Case (1), E has two marked points σ(0), τ (0) ∈ E and one node, and in Case (2), E has one marked point τ (0) ∈ E and two nodes. Figure 4.1: Two families for Smooth case Figure 4.2: Two families for Nodal case Proof. Since the Deligne-Mumford moduli space Mg,n+i is compact, there is a subsequence 39 nk such that [Cn0 ] → [C00 ] for some (g, n + i) curve C00 . Let k C0 π0 (4.4) B0 be a Kuranishi family of C00 = (π 0 )−1 (0) as in Definition 2.1.7. For Smooth case (1), this family comes with two sections σ and τ corresponding to the last two marked points, and also with a forgetful map Φ as in (4.3), which forgets σ and τ and collapses unstable components. For Nodal case (2), this family comes with one section τ corresponding to the last marked point, and also with a forgetful map Φ as in (4.3), which forgets τ and collapses unstable components. Then we can choose b0k → 0 in B 0 and for each k, an identification of Cn0 k with a fiber (π 0 )−1 (b0k ) of C 0 such that qk = σ(b0k ) and rk = τ (b0k ) for Smooth case (1), and rk = τ (b0k ) for Nodal case (2). In any case, the restriction of Φ on C00 is the map collapsing a rational curve E, and is biholomorphic on C00 \ E. Note that in Case (1), E has two marked points σ(0), τ (0) ∈ E and one node, and in Case (2), E has one marked point τ (0) ∈ E and two nodes. (For details, see [ACG] section 10.6 and 10.8) Lemma 4.1.2. Fix a family π : C → B and suppose Ck → C0 in C and p ∈ C0 is a regular node. Also suppose that there is bigger family C 0 of (g, n0 ) curves with n < n0 and forgetful map Φ : C 0 → C. Fix C00 be a fiber of C 0 such that Φ(C00 ) = C0 . Then any node q ∈ C00 with Φ(q) = p is regular. Proof. Since p is regular, there exists a family of (g, l) curves π : C → B with l < n and a forgetful map Φ : C → C such that p := Φ(p) is a regular point. Then composition of forgetful maps Φ ◦ Φ : C 0 → C is also a forgetful map from a family of (g, n0 ) curves to a 40 family of (g, l) curves. Moreover, Φ ◦ Φ(q) = p is a regular point. So q is regular. 4.2 Coordinate expression for Smooth case (1) We identify E with CP 1 by mapping σ(0) to [0 : 1], τ (0) to [1 : 1], and the node of E to [1 : 0]. Then E is covered by two charts W = {[z : 1] : |z| < R} (4.5) and V = {[1 : z 0 ] : |z 0 | < δ} (4.6) for some R, δ with δR > 1 to ensure that W ∪ V covers E. Also assume that σ(0), τ (0) ∈ W . We then identify W with BR by mapping [z : 1] to z, and V with Bδ by mapping [1 : z 0 ] to z0. We will describe coordinate expression in the family C 0 near E ' CP 1 which agrees with coordinate of C near p under forgetful map Φ. Description about coordinates can be found with more details in [ACG] section 10.8. Note that, for any fiber Cb in C, Φ−1 (Cb ) is a two parameter subfamily of the family π 0 , consisting of (Cb , q, r) and its limit case q = r, which is a 1 dimensional subset of nodal family whose fibers look like Cb ∪ CP 1 , parametrized by q which denotes the gluing position. That means, near nodes, the nodal charts center at q and r −q represents how far the smooth fiber is from nodal one. Denote p0 be a nodal point in E ⊂ C00 which maps to p under Φ. Since p is not a node, we can choose local regular chart (x, b) centered at p, so identify neighborhood of p by U × B 41 for some U ⊂ C0 . Since σ and τ are marked sections in C 0 , their images under forgetful map Φ are also sections in C. Note that Φ(σ)(b) = q and Φ(τ )(b) = r. Now translate coordinate x0 = x − q so that Φ(σ) is given by {x0 = 0} and Φ(τ ) is given by {x0 = t} where t = r − q. Choose a homogeneous coordinate [λ : µ] ∈ CP 1 . This gives local coordinate of C 0 near E given by (x0 , t, q, b, [λ : µ]) → (t, q, b) (4.7) with equation x0 µ = tλ. (4.8) Now near node, choose a chart [λ : µ] = [1 : z 0 ] 7→ z 0 with z 0 = µ/λ. Then Equation (4.8) can be written by x0 z 0 = t (4.9) which is the nodal chart near the node. Note that x0 = x − q and t = r − q. This gives the following local chart near the node. (Coordinates near the node) In this chart, the map in (4.3) restricts to Φ U ×V ×U ×B Φ U ×B (x − q, z 0 , q, b) (x, b) π0 π given by π0 π (4.10) U ×U ×B B (r − q, q, b) b |r| where r − q = (x − q)z 0 , and x ∈ U be such that x − q ∈ U and |x − q| ≥ δ . Away from the node, choose a chart [λ : µ] = [z : 1] 7→ z with z = λ/µ. Then Equation (4.8) can be written by λ x0 x−q z= = = . (4.11) µ t r−q 42 Hence we can define Cross-ratio CR(x) = CRq,r (x) = (x−q)/(r−q) which satisfies CR(q) = 0 and CR(r) = 1. This gives the following local chart away from the node. (Coordinates away from the node) Under identification W with BR , coordinates of σ(0) and τ (0) are 0 and 1. The same holds for σ(b) and τ (b), which implies that coordinate for additional marked points qk and rk are 0 and 1 respectively. Hence, as additional parameters, q, r ∈ U determines where the coordinates are 0 and 1. Using Cross-ratio, the map in (4.3) restricts to Φ Φ W ×U ×U ×B U ×B (CR(x), r, q, b) (x, b) π0 π given by π0 π (4.12) U ×U ×B B (r, q, b) b where x−q CR(x) = CRq,r (x) = (4.13) r−q and x ∈ U be such that |CR(x)| ≤ R. Note that, these coordinate systems agree each other. 4.3 Coordinate expression for Nodal case (2) Denote p1 , p2 be two nodal points in E ⊂ C00 such that pi ∈ Ci for i = 1, 2. We identify E with CP 1 by mapping τ (0) to [1 : 1], p1 to [1 : 0], and p2 to [0 : 1]. Then E is covered by three charts V1 = {[1 : z 0 ] : |z 0 | < δ}, (4.14) V2 = {[z : 1] : |z| < δ}, (4.15) 43 Figure 4.3: Coordinate expression for Smooth case and W = {[z : 1] : δ/2 < |z| < 2/δ} (4.16) for some δ < 1. We then identify V1 (and V2 ) with Bδ by mapping [1 : z 0 ] to z 0 (and [z : 1] to z), and W with annulus Aδ = B2/δ \ Bδ/2 by mapping [z : 1] to z. We will describe coordinate expression in the family C 0 near E ' CP 1 which agrees with coordinate of C near the node p under forgetful map Φ. For details, see [ACG] section 10.8. Note that, for nodal fiber C0 in C with node p, Φ−1 (C0 ) is a one parameter subfamily of the family C 0 , consisting of (C0 , r) and its limit case where r = p, which looks like C0 replacing p by exceptional divisor E. Since p is a node, we can choose local nodal chart π(x, y, b̃) = (t, b̃) = b centered at p = (0, 0, 0) with t = xy, so identify neighborhood of p by U1 × U2 × B̃ for some Ui ⊂ Ci 44 for i = 1, 2. Here the projection π : U1 × U2 × B̃ → U × B̃ = B for some U ⊂ C. Since τ is a marked section in C 0 , its image under forgetful map Φ is also a section in C. So using the nodal chart above, Φ(τ )(b) can be writteb by Φ(τ ) = (r, r0 , b̃) with rr0 = t = xy. Choose a homogeneous coordinate [λ : µ] ∈ CP 1 . This gives local coordinate of C 0 near E given by (x, y, r, r0 , b̃, [λ : µ]) → (r, r0 , b̃) (4.17) with equations λr = µx and λy = µr0 . (4.18) Now consider local chart near p1 = [1 : 0]. Choose a chart [λ : µ] = [1 : z 0 ] 7→ z 0 with z 0 = µ/λ. Then Equation (4.18) can be written by xz 0 = r and y = z 0 r0 . (4.19) The first equation describes nodal chart near p1 where x = z 0 = 0. This gives the following local chart near p1 . (Coordinates near node p1 ) In this chart, the map in (4.3) restricts to Φ Φ V1 × U1 × U2 × B̃ U1 × U2 × B̃ (z 0 , x, r0 , b̃) (x, y, b̃) π0 π given by π0 π (4.20) U1 × U2 × B̃ U × B̃ (r, r0 , b̃) (t, b̃) where t = xy = rr0 , y = z 0 r0 , r = xz 0 , and x ∈ U1 be such that z 0 = r/x ∈ V1 ' Bδ . Similarly, consider local chart near p2 = [0 : 1]. Choose a chart [λ : µ] = [z : 1] 7→ z with 45 z = λ/µ. Then Equation (4.18) can be written by x = rz and yz = r0 . (4.21) The second equation describes nodal chart near p2 where y = z = 0. This gives the following local chart near p2 . (Coordinates near node p2 ) In this chart, the map in (4.3) restricts to Φ Φ V2 × U1 × U2 × B̃ U1 × U2 × B̃ (z, r, y, b̃) (x, y, b̃) π0 π given by π0 π (4.22) U1 × U2 × B̃ U × B̃ (r, r0 , b̃) (t, b̃) where t = xy = rr0 , x = rz, r0 = yz, and y ∈ U2 be such that z = r0 /y ∈ V2 ' Bδ . Away from both p1 and p1 , choose a chart [λ : µ] = [z : 1] 7→ z with z = λ/µ. Then the first equation in (4.18) can be written by x z= . (4.23) r Similar with previous subsection, we define Cross-ratio CR(x) = CRp,r (x) = x/r which satisfies CR(r) = 1. This gives the following local chart away from the nodes. (Coordinates away from nodes) In this chart, the map in (4.3) restricts to Φ Φ W × U1 × U2 × B̃ U1 × U2 × B̃ (z, r, r0 , b̃) (x, y, b̃) π0 π given by π0 π (4.24) U1 × U2 × B̃ U × B̃ (r, r0 , b̃) (t, b̃) 46 where x z = CR(x) = CRp,r (x) = (4.25) r and t = xy = rr0 , x = zr, y = r0 /z and x ∈ U1 be such that δ/2 < |CR(x)| < 2/δ. Figure 4.4: Coordinate expression for Nodal case Note that, all these coordinate systems agree each other. 47 Chapter 5 Proof by induction This chapter contains proof of Main Theorem 2.4.6. First we describe exactly where we put marked points with desired properties. Let fk : Ck → X be a sequence of harmonic maps with uniformly bounded energy where Ck are (g, n) curves. By Lemma 2.4.2, fk converges to f0 : C0 → X off the singular set S = Q ∪ N where Q is a set of smooth bubble points and N is a set of nodal bubble points. Denote N ⊂ N be a set of regular nodes. We will put two marked points near points in Q and one marked point near points in N . The very first condition we require for marked points is to converge to points where energy concentrates, hence to build bigger family of curves as in Lemma 4.1.1. In addition, we require 1. at least fixed positive amount of energy is used, or 2. the amount of energy concentration decreases by at least fixed positive amount. Now we focus on the energy density measures on Ck . Set 1 ε̄ = min (ε0 , ε00 , ε000 ). (5.1) 2 48 5.1 Energy concentration at Q If energy concentrates at p ∈ Q (Case 1), we have measure convergence µk → µ∞ + mδp (5.2) as (2.11), where µk = e(f˜k ), µ∞ = e(f0 ) are measures on some open neighborhood U ⊂ C of p with p = 0 and δp is Dirac-delta measure centered at p and m ≥ 2ε̄. For q, r ∈ U , define CRq,r : U → C by x−q CRq,r (x) = . (5.3) r−q Note that CRq,r (q) = 0 and CRq,r (r) = 1. We state marking points lemma that is used to choose marked points. Proof of this lemma will be given in Appendix. Lemma 5.1.1. Suppose µk are sequence of measures on U ⊂ C such that µk → µ∞ + mδp as measures and m ≥ 2ε̄. Then after passing to a subsequence, there exist nested subsets Bk ⊂ U with p = ∩Bk and distinct points qk , rk ∈ Bk such that CRk (Bk ) → C, (5.4) Z νk → m, (5.5) CRk (Bk ) Z νk = ε̄, (5.6) CRk (Bk )\D Z z νk = 0, (5.7) D 49 where CRk = CRqk ,rk , νk = (CRk )∗ µk , and D = {z ∈ C : |z| < 1} is the unit disk. Thus under coordinate change CRk , the measure µk pushes forward to a measure νk on a larger and larger domain CRk (Bk ) ⊂ C such that 1. essentially all of the mass m is captured by νk , 2. all but ε̄ of that mass lies outside the unit disk D, and 3. the center of mass of νk over D is at the origin. Using this lemma, we can describe behavior of energy distribution in new family. Lemma 5.1.2. (Marking points near smooth bubble point) Let p ∈ Q be a smooth bubble point with energy concentration m. Then after passing to a subsequence, there exist unmarked points qk , rk ∈ Ck both converging to p such that 1. Ck0 = (Ck , qk , rk ) → C00 in bigger family C 0 as in Lemma 4.1.1. 2. Denote fk0 = fk ◦ Φ : Ck0 → X. Identify E with CP 1 by mapping σ(0) to [0 : 1], τ (0) to [1 : 1], and the node p0 of E to [1 : 0]. Under the chart [z : 1] 7→ z, Z lim νk = ε̄, (5.8) k→∞ E\D Z z νk = 0 (5.9) D where D = {z : |z| < 1} and νk = e(f˜k0 ) are energy density measures on E. 3. On E, f˜k0 converges to f00 in C 1 away from {p0 } ∪ {qj }j=1,...,l0 where qj ∈ D ⊂ E with 50 energy concentration mj ≥ ε00 . In addition, new node p0 is regular and E(f00 |E ) + X mj = m. (5.10) j=1,...,l0 Proof. First, by Lemma 5.1.1, we can choose points qk , rk ∈ U for each k. Then in regular chart, mark (qk , bk ), (rk , bk ) ∈ Ck as desired marked points. If there is no confusion, we still use qk and rk to refer marked points. Since limk qk = limk rk = p, by Lemma 4.1.1, we have new family of curves C 0 and forgetful map Φ : C 0 → C such that Ck0 = (Ck , qk , rk ) → C00 in C 0 where C00 look like C0 replacing p with a rational curve E, which is isomorphic to CP 1 . Note that Φ(Ck0 ) = Ck and fk0 = fk ◦ Φ : Ck0 → X is also a sequence of harmonic maps. Now identify E with CP 1 by mapping σ(0) to [0 : 1], τ (0) to [1 : 1], and the node p0 of E to [1 : 0] and denote D = {[z : 1] : |z| < 1}. The coordinate away from the node (4.12) is given by (CRq,r (x), r, q, b) which maps to (x, b) under forgetful map Φ. Moreover, this coordinate projected to z = CRq,r (x) under the chart [z : 1] 7→ z. Denote CRk = CRqk ,rk . Then f˜k (x) = fk (x, bk ) = fk0 (CRk (x), rk , qk , bk ) = f˜k0 (CRk (x)) and hence νk = (CRk )∗ e(f˜k ) is the same as e(f˜k0 ). Note that νk can extend to the whole E by Equation (5.4). Since the choice of qk and rk come from Lemma 5.1.1 and cross ratio is conformally invariant, Equations (5.8) and (5.9) come from Equations (5.6) and (5.7). By applying Lemma 2.4.2 again, there is a subsequence and a finite set of bubble points {q1 , . . . , ql } ⊂ E \ {p0 } such that after passing to a subsequence, νk → e(f00 ) + P j mj δqj on E \ {p0 } with mj ≥ ε00 . Here f00 : C00 → X is a limit of fk0 . By Equation (5.8), qj ∈ D. 51 Denote m∞ the amount of energy concentration at p0 , then we have e(f00 )(E) + X mj + m∞ = m. (5.11) j Finally, we will show that m∞ = 0. Define B 0 = {(x, bk ) ∈ Ck : x ∈ Bk }. For any compact set K ⊂⊂ E \ {p0 }, define K 0 = {(CRk (x), rk , qk , bk ) ∈ Ck0 : [CRk (x) : 1] ∈ K} using coordinate (4.12). For simplicity, denote the restriction of forgetful map on Ck0 by Φ = Φ|C 0 . It is enough to show that for any ε > 0, there is some K such that k m∞ ≤ m − e(f00 )(K) − mj = lim (µk (Bk ) − νk (K)) = lim E(fk0 , Φ−1 (B 0 ) \ K 0 ) < ε. X k→∞ k→∞ j Fix ε > 0. Let δ > 0 to be determined and let K = {[1 : z 0 ] : |z 0 | ≥ δ}. We will show that Φ−1 (B 0 ) \ K 0 ⊂ Ck0 ∩ B(p0 , δ) for all k sufficiently large. First as- sume δ small enough such that Φ−1 (B 0 ) \ K 0 lies in the coordinate (4.10). Pick q = (x − qk , z 0 , qk , bk ) ∈ Φ−1 (B 0 ) \ K 0 such that z 0 = (rk − qk )/(x − qk ). Note that because x, qk , rk ∈ Bk , |x − qk |, |qk |, |bk | ≤ δ for all k sufficiently large. Moreover, q ∈/ K 0 means [CRk (x) : 1] ∈ / K, which implies |1/CRk (x)| = |z 0 | < δ for all k sufficiently large. Therefore q ∈ Ck0 ∩ B(p0 , δ) as desired. Note that p0 is regular by definition. By Equation (5.8), E(fk0 , Φ−1 (B 0 ) \ K 0 ) ≤ ε̄ ≤ ε000 /2. Hence, for δ small enough, we have E(fk0 , Ck0 ∩ B(p0 , δ)) < ε000 . Therefore, by Lemma 2.4.5, there is δ > 0 such that limk→∞ E(fk0 , Ck0 ∩ B(p0 , δ)) < ε. So m∞ = 0 and this proves the lemma. 52 5.2 Energy concentration at N If energy concentrates at p ∈ N (Case 2), first consider δ-neighborhood Ck ∩ B(p, δ) = {x, y ∈ C2 : xy = tk , |x|, |y| ≤ δ} of p. Then we have energy convergence E(fk , Ck ∩ B(p, δ)) → E(f0 , C0 ∩ B(p, δ)) + m. (5.12) Then by Lemma 2.4.5, m ≥ ε000 ≥ 2ε̄. Let us transform the domain into the disk so that the above argument can be applied. The domain Ck ∩B(p, δ) can be projected down to the annulus Ak,δ := Bδ \Bt /δ by the projection k π1 (x, y) = x. Note that C0 ∩ B(p, δ) is the union of two δ-balls, say Bδ = B(p, δ)|y=0 and y Bδ = B(p, δ)|x=0 . Then define extended push forward energy density measure µk over U := Bδ by µk = (π1 )∗ e(fk ) on Ak,δ and µk = 0 on Bt /δ . Furthermore, µk → µ∞ + (m + Eδ )δp . k y Here µ∞ = (π1 )∗ e(f0 ) on Bδ , Eδ = E(f0 , Bδ ) and δ0 is Dirac-delta measure centered at p and m ≥ 2ε̄. By choosing δ small, we can make Eδ as small as we want. Therefore, without loss of generality, we just denote m instead of m + Eδ and consider µk → µ∞ + mδp . (5.13) The following lemma is similar as Lemma 5.1.1, and proof will be given in Appendix. Lemma 5.2.1. Suppose µk are sequence of measures on U ⊂ C such that µk → µ∞ + mδp as measures and m ≥ 2ε̄. Then after passing to a subsequence, there exist nested subsets 53 Bk ⊂ U with p = ∩Bk and rk ∈ Bk such that CRk (Bk ) → C, (5.14) Z νk → m, (5.15) CRk (Bk ) Z νk = ε̄, (5.16) CRk (Bk )\D where CRk = CRp,rk , νk = (CRk )∗ µk , and D = {z ∈ C : |z| < 1} is the unit disk. Using this lemma, we can describe behavior of energy distribution in new family. Lemma 5.2.2. (Marking points near regular nodal bubble point) Let p ∈ N be a regular nodal bubble point with energy concentration m. Then after passing to a subsequence, there exist unmarked points rk ∈ Ck converging to p such that 1. Ck0 = (Ck , rk ) → C00 in bigger family C 0 as in Lemma 4.1.1. 2. Denote fk0 = fk ◦ Φ : Ck0 → X. Identify E with CP 1 by mapping one node p2 of E to [0 : 1], τ (0) to [1 : 1], and the other node p1 of E to [1 : 0]. Under the chart [z : 1] 7→ z, Z lim νk = ε̄ (5.17) k→∞ E\D where D = {z : |z| < 1} and νk = e(f˜k0 ) are energy density measures on E. 3. On E, f˜k0 converges to f00 in C 1 away from {p1 }∪{p2 }∪{qj }j=1,...,l0 where qj ∈ D ⊂ E with qj 6= p2 and with energy concentration mj ≥ ε00 . In addition, new nodes p1 , p2 are regular and E(f00 |E ) + X mj + m0 = m (5.18) j=1,...,l0 54 where m0 is the energy concentration at p2 ∈ E and m0 ≥ ε000 . Proof. By Lemma 5.2.1, we can choose a point rk ∈ U for each k. Then in nodal chart, mark (rk , tk /rk , b̃k ) ∈ Ck as desired marked points. Here we need to check tk /rk → 0. Suppose that there is c > 0 such that |tk /rk | ≥ c for all k. Since µk = 0 on Bt /δ and Bcr /δ ⊂ Bt /δ , µk = 0 on Bcr /δ for all k. Recall k k k k that CRk (x) = CRp,rk (x) = x/rk . Choose δ small enough such that CRk−1 (D) = Brk ⊂ Bcr /δ for all k. Therefore µk = 0 on CRk−1 (D). Now Equation (5.16) can be rewritten k R as dµk = µk (Bk ) = ε̄, which contradicts µk (Bk ) → m ≥ 2ε̄. Therefore Bk \CR−1 (D) k tk /rk → 0. If there is no confusion, we still use rk to refer marked points. Since limk rk = p, by Lemma 4.1.1, we have new family of curves C 0 and forgetful map Φ : C 0 → C such that Ck0 = (Ck , rk ) → C00 in C 0 . Note that Φ(Ck0 ) = Ck and fk0 = fk ◦ Φ : Ck0 → X is also a sequence of harmonic maps. Now identify E with CP 1 by mapping one node p2 of E to [0 : 1], τ (0) to [1 : 1], and the other node p1 of E to [1 : 0] and denote D = {[y : 1] : |y| < 1}. The coordinate (4.24) away from the nodes is given by (CRp,r (x), r, r0 , b̃) which maps to (x, y, b̃) under forgetful map Φ. Moreover, this coordinate projected to z = CRp,r (x) under the chart [z : 1] 7→ z. Denote CRk = CRp,rk . Then (π1 )∗ fk (x) = fk (x, tk /x, b̃k ) = fk0 (CRk (x), rk , tk /rk , b̃k ) = f˜k0 (CRk (x)) and hence νk = (CRk )∗ (π1 )∗ e(fk ) is the same as e(f˜k0 ). Note that νk can extend to the whole E by Equation (5.14). Since the choice of rk come from Lemma 5.2.1 and cross ratio is conformally invariant, Equation (5.17) comes from Equation (5.16). 55 By applying Lemma 2.4.2 again, there is a subsequence and a finite set of bubble points {q1 , . . . , ql } ⊂ E \ {p1 , p2 } such that after passing to a subsequence, νk → e(f00 ) + P j mj δqj on E \ {p1 , p2 } with mj ≥ ε00 . Here f00 : C00 → X is a limit of fk0 . By Equation (5.6), qj ∈ D and qj 6= p2 . Denote m0 and m∞ the amount of energy concentration at p2 and at p1 respectively. Since p is regular node, p1 and p2 are also regular by Lemma 4.1.2. Hence m0 , m∞ are either zero or at least ε000 by Lemma 2.4.5. Now we have e(f00 )(E) + X mj + m0 + m∞ = m. (5.19) j Finally, we will show that m∞ = 0. Define B 0 = {(x, y, b̃k ) ∈ Ck ∩ B(p, δ) : xy = tk , x ∈ Bk }. For any compact set K ⊂⊂ E \ {p1 , p2 }, define K 0 = {(CRk (x), rk , tk /rk , b̃k ) ∈ Ck0 : [CRk (x) : 1] ∈ K} using coordinate (4.24). For simplicity, denote the restriction of forgetful map on Ck0 by Φ = Φ|C 0 . Then we have k m0 + m∞ ≤ m − e(f00 )(K) − mj ≤ lim (µk (Bk ) − νk (K)) = lim E(fk0 , Φ−1 (B 0 ) \ K 0 ). X k→∞ k→∞ j Let 0 < δ 0 < δ to be determined and let K = {[z : 1] : δ 0 ≤ |z| ≤ 1/δ 0 }. Assume δ 0 small enough such that Φ−1 (B 0 ) \ K 0 lies in either coordinate (4.20) or coordinate (4.22). Then Φ−1 (B 0 ) \ K 0 = K1 ∪ K2 where K1 := {q = (z 0 , x, tk /rk , b̃k ) ∈ Φ−1 (B 0 ) \ K 0 : |z 0 | < δ 0 } K2 := {q = (z, rk , tk /x, b̃k ) ∈ Φ−1 (B 0 ) \ K 0 : |z| < δ 0 }. Note that m0 ≤ limk→∞ E(fk0 , K2 ) and m∞ ≤ limk→∞ E(fk0 , K1 ). We will show that K1 ⊂ Ck0 ∩ B(p1 , δ 0 ) and K2 ⊂ Ck0 ∩ B(p2 , δ) for all k sufficiently large. 56 Let q ∈ K1 . Note that because x, rk ∈ Bk , |x|, |tk /rk |, |b̃k | ≤ δ 0 for all k sufficiently large. Moreover, |z 0 | < δ 0 by definition of K1 . Therefore q ∈ Ck0 ∩ B(p1 , δ 0 ) as desired. Next, let q ∈ K2 . As above, |rk |, |b̃k | ≤ δ 0 for all k sufficiently large. We also have |tk /x| = |y| ≤ δ. Moreover, |z| < δ 0 by definition of K2 . Therefore q ∈ Ck0 ∩ B(p2 , δ) as desired. Now it is enough to show that for any ε > 0, there is δ 0 such that m∞ ≤ E(fk0 , Ck0 ∩ B(p1 , δ 0 )) ≤ ε. (5.20) Fix ε > 0. By Equation (5.6), E(fk0 , K1 ) ≤ ε̄ ≤ ε000 /2. Hence, for δ 0 small enough, we have E(fk0 , Ck0 ∩ B(p1 , δ 0 )) < ε000 . Therefore, by Lemma 2.4.5, there is δ 0 > 0 such that limk→∞ E(fk0 , Ck0 ∩ B(p1 , δ 0 )) < ε. So m∞ = 0 and this proves the lemma. 5.3 Completion of the proof In this section we prove Main Theorem 2.4.6 and its special cases, Corollaries 2.4.7 and 2.4.8. First we define the residual energy. Definition 5.3.1. Suppose fk : Ck → X be a sequence of maps with uniformly bounded energy that converges to f0 : C0 → X off the set S = Q ∪ N , where Q = {p1 , . . . , pl } is a set of smooth bubble points and N is a set of nodal bubble points. Let N = {q1 , . . . , qn } be a set of regular nodes that energy concentrates. Define residual energy, denoted by RE, by RE = lim E(fk ) − E(f0 ) − lε̄ − nε̄/2. (5.21) k→∞ 57 If we denote mi be energy concentration at pi and m0j be energy concentration at qj , then since mi , m0j ≥ 2ε̄, we have l n (m0j − ε̄/2) ≥ lε̄ + 3mε̄/2. X X RE = (mi − ε̄) + (5.22) i=1 j=1 Recall the statement of Main Theorem: Theorem 2.4.6. (Main Theorem) Suppose fk : Ck → X be a sequence of harmonic maps with uniformly bounded energy defined on smooth (g, n) curves. Then there is a subsequence nk and a way of marking points Pk on Ck such that cor- responding sequence fnk : Cn0 = (Cnk , Pnk ) → X converges to some f0 : C0 → X off the k singular set S in C 1 where all points in S are non-regular nodal points. Furthermore, f0 is harmonic on closure of each component of C0 \ S separately. Proof. If there is a subsequence nk and a finite set of points Pk on Ck such that Cn0 = k (Cnk , Pnk ) → C00 for some C00 and corresponding residual energy RE = 0, then there is no energy concentration points except non-regular nodes and we are done. Now suppose RE > 0 for any subsequence nk and any set of marking points Pk such that Cn0 converges. That means, energy concentration occurs at either p ∈ Q or at p ∈ N . k Case 1: There is energy concentration m at p ∈ Q. By Lemma 5.1.2, after passing to a subsequence, we can add two marked points qk , rk ∈ P0 Ck0 such that E(f00 |E ) + lj=1 mj = m, where mj ≥ 2ε̄. In the new family, RE 0 = lim E(fk0 ) − E(f00 ) − (l − 1 + l0 )ε̄ − nε̄/2. (5.23) k→∞ 58 Note that n does not change because the new node is a regular node with no energy concen- tration. Then the difference of new residual energy from old one is RE 0 − RE = −(l0 − 1)ε̄ − E(f00 |E ). (5.24) If E(f00 |E ) > 0, then since E(f00 |E ) ≥ 2ε̄, RE 0 ≤ RE − ε̄. If E(f00 |E ) = 0 and l0 ≥ 2, then RE 0 ≤ RE − ε̄. Finally, if E(f00 |E ) = 0 and l0 ≤ 1, we know l0 = 1 because of the energy P0 identity E(f00 |E ) + lj=1 mj = m. Note that from (5.9), the location of the bubble on E is [0 : 1]. But then (5.8) implies that energy of amount of ε̄ on a subset of E \ D can not be used for the bubble, hence E(f00 |E ) ≥ ε̄. This contradicts to the assumption E(f00 |E ) = 0, so this case is impossible. Hence, in any case, RE 0 ≤ RE − ε̄. Case 2: There is energy concentration m at a regular node p ∈ N . By Lemma 5.2.2, we can add one marked point rk ∈ Cn0 and a subsequence n0k of nk k 0 such that E(f00 |E ) + lj=1 mj + m∞ = m, where ν0 = e(f00 ), mj ≥ 2ε̄, and m∞ is either P zero or at least 2ε̄. In the new family, RE 0 = lim E(fk0 ) − E(f00 ) − (l + l0 )ε̄ − n0 ε̄/2 (5.25) k→∞ where n0 is the number of new regular nodes that energy concentrates. Note that n0 = n if m∞ 6= 0 or n0 = n − 1 if m∞ = 0, so the difference of new residual energy from old one is RE 0 − RE ≤ −l0 ε̄ + ε̄/2 − E(f00 |E ). (5.26) 59 If E(f00 |E ) > 0, then since E(f00 |E ) ≥ 2ε̄, RE 0 ≤ RE − ε̄. If E(f00 |E ) = 0 and l0 ≥ 1, then RE 0 ≤ RE − ε̄/2. Finally, if E(f00 |E ) = 0 and l0 = 0, then m∞ = m and all energy concentrates at [0 : 1]. But from (5.6), energy of amount of ε̄ on a subset of E \ D can not concentrate at [0 : 1], which contradicts to E(f00 |E ) = 0. So this case is impossible. Hence, in any case, RE 0 ≤ RE − ε̄/2. In conclusion, if RE > 0, we mark either two points near a bubble point or one point near regular nodal point, and make RE 0 ≤ RE − ε̄/2. Since RE is finite, this process should stop when energy concentrates only at non-regular nodes. That means, there is a subsequence nk and a way of marking points Pk on Ck such that Cn0 = (Cnk , Pnk ) → C00 for some k C00 and there is no energy concentration point other than non-regular nodes. Therefore, corresponding sequence fnk : Cn0 = (Cnk , Pnk ) → X converges to f0 : C0 → X off the k singular set S in C 1 , for some marked nodal curve C0 and limit map f0 and S only consists of non-regular nodes. The harmonicity of f0 comes from Lemma 2.4.2. Next, consider its corollaries: Corollary 2.4.7. Suppose fk : Ck → X be a sequence of harmonic maps with uniformly bounded energy defined on smooth (g, n) curves. Also assume Ck → C0 in a family C and all nodes in C0 are regular. Then there is a subsequence nk and a way of marking points Pk on Ck such that corre- sponding sequence fnk : Cn0 = (Cnk , Pnk ) → X converges to some f0 : C00 → X in C 1 where k Cn0 → C00 in bigger family C 0 . That means, the singular set S in Theorem 2.4.6 is empty. k Furthermore, f0 is harmonic on each irreducible component of C0 , the energy identity holds, and the image of f0 is connected. Proof. The only thing we need to show is that all nodes of C00 are regular. Pick p0 be a node 60 in C00 . Since forgetful map Φ : C 0 → C maps C00 to C0 , Φ(p0 ) ∈ C0 is either regular point or nodal point. If Φ(p0 ) is regular point, p0 is regular node by definition. If Φ(p0 ) is nodal point, by assumption, it is regular nodal point. So by Lemma 4.1.2, p0 is regular node. Hence by Theorem 2.4.6, there is a subsequence nk and a way of marking points Pk on Ck such that fnk : Cn0 = (Cnk , Pnk ) → X converges to some f0 : C00 → X in C 1 with k singular set S and S is empty. Corollary 2.4.8. Let Σ be a smooth Riemann surface with genus g and suppose fk : Σ → X be a sequence of harmonic maps with uniformly bounded energy. Then there is a subsequence nk , a way of marking points Pk on Σ, a marked nodal curve C0 , and a limit map f0 : C0 → X such that corresponding sequence fnk : Cnk = (Σ, Pnk ) → X converges to f0 in C 1 . Furthermore, f0 is harmonic on each irreducible component of C0 , the energy identity holds, and its image is connected. Proof. Let Ck = Σ, then C0 = Σ which do not have any node. The corollary then follows from Corollary 2.4.7. 61 APPENDIX 62 Appendix In the appendix we describe the proofs of Lemmas 5.1.1 and 5.2.1. Assume the convergence of measures µk → µ∞ + mδp (A.1) where µk , µ∞ are measures on U ⊂ C, δp is Dirac-delta measure at the origin and m ≥ 2ε̄. Throughout this chapter, B(x, r) ⊂ U denotes a ball of radius r in U centered at x ∈ U . Choose δ0 > 0 such that Z ε̄ dµ∞ ≤ . (A.2) B(0,δ0 ) 4 From (A.1), given any ε < ε̄, δ < δ0 , there is a subsequence µnk of µk such that Z dµnk − dµ∞ − mδp < ε (A.3) B(0,δ) and Z dµnk − dµ∞ < ε (A.4) B(0,δ0 )\B(0,δ) for all k. Now extract subsequences as follows: Pick εk → 0 with εk ≤ εk−1 /2. Pick δk → 0 with δk ≤ δk−1 /2 such that Z dµ∞ ≤ εk . (A.5) B(0,δk ) For simplicity, denote Bk = B(0, δk ) for k = 0, 1, 2, · · · . For (ε1 , δ1 ), there exist a subsequence µ1k of µk such that (A.3) and (A.4) hold with (ε1 , δ1 ). For (ε2 , δ2 ), there exist 63 further subsequence µ2k of µ1k such that (A.3) and (A.4) hold with (ε1 , δ1 ) and (ε2 , δ2 ). Keep going and choose diagonal of above, say µkk . Finally rename µk = µ2k 2k . Then, given any k, for all 1 ≤ m ≤ 2k, we have Z dµk − dµ∞ − mδp < εm (A.6) Bm and Z dµk − dµ∞ < εm . (A.7) B0 \Bm Now for following lemmas, we will fix k and denote µk by simply µ. We first clarify which assumption we will use. Assumption A.1. Assume µ is a smooth finite mass measure on a bounded set U ∈ C. Note that, by choosing k large enough we have 1. 2εk + 2ε2k < ε̄. 2. 3δ2k−1 < δk . 3. E := µ(Bk ) > m − 2εk > ε̄ by (A.6). 4. (E − ε̄ − 2εk − 2ε2k )/2 > 8(εk + ε2k ). Definition A.2. Given q ∈ Bk and t ∈ (0, 1), define rt = q + t/(1 − t). Also define cross ratio Rq,t (w) : U → C by (x − q) 1−t Rq,t (x) = = (x − q) = (t−1 − 1)(x − q). (A.8) (rt − q) t 64 Note that for fixed q, • as t → 0, rt → q and Rq,t (x) → ∞ for all x 6= q. • as t → 1, rt → ∞ and Rq,t (x) → 0 for all x. Lemma A.3. Let µ be as in Assumption A.1. Given q ∈ Bk , there exists a unique t = tq ∈ (0, 1) such that Z (Rq,tq )∗ dµ = ε̄ (A.9) Rq,tq (Bk )\D where D is unit disk in C. Proof. Define a continuous function f (t) : (0, 1) → [0, ∞) by Z Z f (t) = (Rq,t )∗ dµ = dµ (A.10) Rq,tq (Bk )\D At where At = Bk \ Rq,t −1 (D) = {x ∈ Bk : |Rq,t (x)| > 1}. Now for x 6= q, ∂ |Rq,t (x)| = −t−2 |x − q| < 0 ∂t so {At } is a family of sets strictly descending on t hence f (t) is strictly decreasing. Note that Z Z lim f (t) = lim dµ = dµ = E > ε̄ t→0 t→0 A Bk \{q} t Z Z lim f (t) = lim dµ = dµ = 0 < ε̄ t→1 t→1 A ∅ t hence there exists a unique tq such that f (tq ) = ε̄. Definition A.4. For simplicity, denote Rq = Rq,tq . 65 Lemma A.5. B2k 6⊂ Rq−1 (D) for any q ∈ Bk . Proof. From (A.6), we have Z dµk − dµ∞ − mδp < εk Bk Z Z dµk < m + dµ∞ + εk < m + 2εk . Bk Bk Also, we have Z dµk − dµ∞ − mδp < ε2k B2k Z Z dµk > m − dµ∞ − ε2k > m − 2ε2k . B2k B2k Therefore, Z dµk < 2εk + 2ε2k < ε̄. (A.11) Bk \B2k If B2k ⊂ Rq−1 (D), then Z Z ε̄ = dµk < dµk < ε̄ (A.12) Bk \Rq−1 (D) Bk \B2k which is a contradiction. Hence we know B2k 6⊂ Rq−1 (D) for any q ∈ Bk . Lemma A.6. For any given q ∈ Bk , |q| + δ2k tq ≤ . (A.13) 1 + |q| + δ2k 66 Proof. By Lemma A.5, there exists x0 ∈ B2k such that x0 6∈ Rq−1 (D). Hence, 1 − tq 1< |x0 − q| (A.14) tq tq < |x0 − q| ≤ |q| + δ2k . (A.15) 1 − tq Solving this inequality for tq proves the lemma. Lemma A.7. Suppose |q| ≤ δ2k−1 . Then for any x ∈ Rq−1 (D), we have |x| ≤ 3δ2k−1 . (A.16) Moreover, Rq−1 (D) ⊂ Bk . Proof. Let x ∈ Rq−1 (D). Then from Equation (A.15), we have 1 − tq |x − q| ≤ 1 tq tq |x − q| ≤ 1 − tq tq |x| ≤ |q| + ≤ 2|q| + δ2k ≤ 3δ2k−1 . 1 − tq Lemma A.8. The assignment q 7→ tq is continuous on q ∈ Bk . Proof. Fix q. Denote f (t) = t/(1 − t) and its inverse g(s) = s/(s + 1), and let sq = f (tq ). For any ε > 0, since g is continuous, there is δ > 0 such that if s satisfies |s − sq | < δ, then |g(s) − g(sq )| < ε. Choose q 0 ∈ Bk be such that |q − q 0 | < δ. Now it is enough to show that |tq − tq0 | ≤ ε. 67 Claim : If q, q 0 ∈ Bk satisfy |q − q 0 | < δ and t, t0 satisfy f (t) = f (t0 ) + δ, then Bk ∩ Rq0 ,t0 −1 (D) ⊂ Bk ∩ Rq,t −1 (D). (A.17) Proof of Claim. Suppose x ∈ Bk ∩ Rq0 ,t0 −1 (D). Then 1 − t0 |x − q 0 | ≤ 1 t0 t0 |x − q 0 | ≤ 0 = f (t0 ) 1−t t |x − q| ≤ f (t0 ) + |q − q 0 | ≤ f (t0 ) + δ = f (t) = 1−t 1−t |x − q| ≤ 1. t Hence x ∈ Bk ∩ Rq,t −1 (D) and claim is proved. Now let t1 < t2 such that f (tx ) = f (t1 ) + δ = f (t2 ) − δ. Then by the claim, Bk ∩ Rq0 ,t −1 (D) ⊂ Bk ∩ Rq,tq −1 (D) ⊂ Bk ∩ Rq0 ,t −1 (D) hence t1 < tq0 < t2 from the definition 1 2 of tq0 . Now we get |f (tq ) − f (tq0 )| = |sq − sq0 | < δ and so |g(sq ) − g(sq0 )| = |tq − tq0 | < ε. Definition A.9. Denote νq = (Rq )∗ µ. Define F : Bk → C by Z F (q) = zdνq (z). (A.18) D Proposition A.10. F(q) in Definition A.9 is continuous on q ∈ Bk . Proof. From Lemma A.8 and Equation (A.8), it is obvious that Rq = CRq,tq is continuous on q ∈ Bk . Hence push-forward measure νq = (Rq )∗ µ is also continuous on q, and F (q) is also continuous on q. 68 Proposition A.11. Let F (q) be in Definition A.9. There exists qk ∈ B2k−1 such that F (qk ) = 0. This proposition is less obvious because if F (q) = (q + 2δk )/3, then |F (q)| ≥ δk /3 > 0. To avoid this case, we need the following lemma. Lemma A.12. Let F (q) be in Definition A.9. For any given q ∈ ∂B2k−1 ,   F (q) Re > 0. (A.19) −q Proof. 1 − tq Z Z Z F (q) = zdνq (z) = Rq (x)dµ(x) = (x − q)dµ(x). (A.20) D Rq−1 (D) tq Rq−1 (D) Denote f (x) = x − q. If x ∈ B2k , then for u = x/q,   f (x) Re = Re(1 − u) > 0 f (0) because |u| ≤ 1/2. Define sets A, B, C by A := {x ∈ Rq−1 (D) : x ∈ B2k }     −1 f (x) B := x ∈ Rq (D) : x 6∈ A and Re ≥0 f (0)     −1 f (x) C := x ∈ Rq (D) : Re <0 . f (0) Note that µ(A) + µ(B) + µ(C) = µ(Rq−1 (D)) = E − ε̄ and µ(B) + µ(C) ≤ µ(Bk \ B2k ) < 69 2εk + 2ε2k by Equation (A.11). So we have µ(A) = E − ε̄ − µ(B) − µ(C) ≥ E − ε̄ − 2εk − 2ε2k and µ(C) ≤ 2εk + 2ε2k .   Z   F (q) tq f (x) Re = Re dµ(x) −q 1 − tq A∪B∪C f (0) Z   Z   f (x) f (x) ≥ Re dµ(x) + Re dµ(x) A f (0) C f (0) δ2k−1 − δ2k Z Z δ2k−1 + 3δ2k−1 ≥ dµ − dµ A δ2k−1 C δ2k−1 1 1 ≥ µ(A) − 4µ(C) ≥ (E − ε̄ − 2εk − 2ε2k ) − 8(εk + ε2k ) > 0. 2 2 Hence we get Re (−F (q)/q) > 0. Proof. (Proof of Proposition A.11) Note that by Lemma A.12, F (∂B2k−1 ) is a closed curve with nonzero index. By considering homotopy between ∂B2k−1 and {0}, we can show that F (∂B2k−1 ) is contractible. So there exists x ∈ B2k−1 such that F (q) = 0. Now we go back to original sequence with subscript k. For qk in Proposition A.11, denote tk = tqk , rk = rtk and CRk = CRqk ,rk := Rqk . To finalize the proof of Lemmas 5.1.1 and 5.2.1, we need rk ∈ Bk and CRk (Bk ) → C. Lemma A.13. rk ∈ Bk . Also, CRk (Bk ) → C as k → ∞. Proof. tk |rk | ≤ |qk | + ≤ 2|qk | + δ2k ≤ 3δ2k−1 ≤ δk 1 − tk which proves the first. 70 To show the second, it is enough to show that for any R > 0, for all k large enough, CRk−1 (DR ) ⊂ Bk where DR ⊂ C is a disk of radius R. Fix R > 0 and choose x ∈ CRk−1 (DR ). Then we have tk |x| ≤ |qk | + R ≤ δ2k−1 + R(δ2k−1 + δ2k ) ≤ (1 + 2R)2−k+1 δk . 1 − tk Now choose k large enough so that (1 + 2R)2−k+1 ≤ 1. Now we are ready to prove Lemmas 5.1.1 and 5.2.1. Recall the statements: Lemma 5.1.1. Suppose µk are sequence of measures on U ⊂ C such that µk → µ∞ + mδp as measures and m ≥ 2ε̄. Then after passing to a subsequence, there exist nested subsets Bk ⊂ U with p = ∩Bk and distinct points qk , rk ∈ Bk such that CRk (Bk ) → C, (5.4) Z νk → m, (5.5) CRk (Bk ) Z νk = ε̄, (5.6) CRk (Bk )\D Z z νk = 0, (5.7) D where CRk = CRqk ,rk , νk = (CRk )∗ µk , and D = {z ∈ C : |z| < 1} is the unit disk. Proof. Choose qk as in Proposition A.11 and rk as in Lemma A.3 . Then Equations (5.6) and (5.7) follows. Also, Equation (5.5) comes from Equation (A.6). Furthermore, K ⊂ CRk (Bk ) comes from Lemma A.13. Lemma 5.2.1. Suppose µk are sequence of measures on U ⊂ C such that µk → µ∞ + mδp as measures and m ≥ 2ε̄. Then after passing to a subsequence, there exist nested subsets 71 Bk ⊂ U with p = ∩Bk and rk ∈ Bk such that CRk (Bk ) → C, (5.14) Z νk → m, (5.15) CRk (Bk ) Z νk = ε̄, (5.16) CRk (Bk )\D where CRk = CRp,rk , νk = (CRk )∗ µk , and D = {z ∈ C : |z| < 1} is the unit disk. Proof. Choose qk = p and rk as in Lemma A.3. Then Equation (5.16) comes from Lemma A.3. 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