THE-Sta t :3 00‘ This is to certify that the dissertation entitled The Moduli Space of Special Lagrangian Submanifolds presented by Sema Salur has been accepted towards fulfillment of the requirements for Ph .0. degree in Ma‘l’hema‘fics @flaiaue Major professor Date 12/15/00 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Michigan State Unlvorslty PLACE IN RETURN BOX to remove thi TO AVOID FINES return on MAY BE RECALLED with earlier 5 checkout from your record. or before date due. due date if requested. DATE DUE DATE DUE DATE DUE JUN182 DB 11100 W.“ The Moduli Space of Special Lagrangian Submanifolds By Sema Salur A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2000 ABSTRACT The Moduli Space of Special Lagrangian Submanifolds By Sema Salur In this thesis we study the deformation theory of special Lagrangian subman— ifolds and the singularities of the local moduli space. We show that the moduli space of all infinitesimal special Lagrangian deformations of a smooth, compact, orientable special Lagrangian L in a symplectic manifold with non-integrable al- most complex structure is a smooth manifold and its dimension is equal to the dimension of 711(L), the space of harmonic 1-forms on L. ACKNOWLEDGMENTS I would like to express my sincere gratitude to Gang Tian, my dissertation advisor for all his help, guidance and encouragement. I would also like to thank all my guidance committee members David Blair, Ronald Fintushel, John McCarthy, Tom Parker, and Zhengfang Zhou for their help. I am especially grateful to Peter Ozsvath for very useful conversations for my research. Finally, thanks to Selman Akbulut for being the biggest support from the beginning. iii TABLE OF CONTENTS 1 Introduction 1.1 Calibrated Geometries ......................... 1.1.1 Complex Submanifolds of Kahler Manifolds ......... 1.1.2 Calibrations ........................... 1.1.3 Special Lagrangian Geometries ................ 1.2 Calabi-Yau Manifolds and Strominger-Yau-Zaslow Conjecture 1.3 Organization .............................. 2 Deformations of Special Lagrangian Submanifolds 2.1 Deformations in a Calabi-Yau Manifold ................ 2.2 A New Interpretation of Special Lagrangian Submanifolds ..... 2.3 Deformation Theory in a Symplectic Manifold ............ BIBLIOGRAPHY iv 14 14 17 20 31 CHAPTER 1 Introduction The purpose of this chapter is to develop some motivation for the study of special Lagrangian submanifolds and to give some insight into the geometry involved. We will first give the basic definitions in calibrated geometries and study the complex submanifolds of Kahler manifolds as an example. We will then discuss the general concept of special Lagrangian geometries with possible applications. 1.1 Calibrated Geometries Given a real valued function f (:13) : R —> R we can describe the local minimum points of f by finding the critical points (a:| f’ (at) = 0) and applying some derivative tests. By comparing the local minimum points we can then describe the global minimum points. Let X n be a Riemannian manifold. We can also study a similar problem in the space of immersions of M into X where f is now defined as a volume functional. Given a E Hk(X,Z) define the set 71 = {M: compact, oriented submanifolds of X | [M] = a} and the volume functional V : H —> R such that V(M) 2/ dvolM. Our goal in this chapter is to study” the global minimum points of this functional. The motivation behind this is that a calibrated submanifold is volume minimizing in its homology class. Therefore in order to understand the geometry of this special class of submanifolds we should first understand the geometry of V(M) and in particular the global minimum points of V. 1.1.1 Complex Submanifolds of Kéihler Manifolds In this section we study the global minimum points of the volume functional of a Kahler manifold and show that a complex submanifold has least volume in its homology class by proving the Wirtinger’s Inequality and the Federer’s Argument. First we will give some definitions. DEFINITION 1.1 : A symplectic structure on a manifold X 2" is a nondegenerate closed 2-form w E 92(X). A smooth manifold X 2" with a symplectic structure is called a symplectic manifold. (e. 9 R2", 32) Nondegeneracy means that each tangent space (TqX, wq) at any point q E X is a symplectic vector space. REMARK 1.1 : The symplectic manifold X is necessarily of even dimension be- cause the n-fold wedge product w A w /\ /\ w never vanishes. This also implies that X is orientable. DEFINITION 1.2 : An almost complex structure on a 2n dimensional real manifold is a complex structure J on the tangent bundle TX. DEFINITION 1.3 : An almost complex structure J on a symplectic manifold is called integrable if and only if it is covariant constant with respect to the Levi- Civita connection of the associated metric g J. DEFINITION 1.4 : A Kc'ihler manifold is a symplectic manifold (X,w) with an integrable almost complex structure J. (e.g R2", CP") Recall that our goal is to study the volume minimizing submanifolds of a Kahler manifold. For this we need two important tools, namely the Wirtinger’s Inequality and the Federer’s Argument. Theorem 1.1 : (Wirtinger’s Inequality) Let X 2" be a Kiihler manifold and M 2’" be a real, oriented submanifold of X. Then for all p E M, -“’,;,'3.IT,.M S dellrpM with equality if and only if TpM is a complex subspace of TpX . Proof: For any two unit vectors Y, W E TPX, we can show that w(Y, W)2 = < Y, JW >2 (compatibility of w and the metric) g |Y|2|JW|2 (by Cauchy-Schwartz inequality) = |Y|2|W|2 = 1 (J preserves the length of the vectors) and the inequality above will be an equality iff Y = :l:.I W, in other words iff Y and W span a complex space. By linear algebra we can show that there exists an oriented, orthonormal basis e1, ...82m of Tp(M) such that oz can be written as follows: m w = 2.54 )‘iw2i—l /\ w2i where A,- = w(e2,-_1,e2,-) for i = 1, ..,m and w1,...,w2m are the dual one forms to 61, ...€2m. With a simple calculation one can show that w'" = (m!))\1...)\mw1/\ /\ (412m. lwml = (m!)w1 A szm iff |/\1.../\m| = 1. Since we showed earlier that w(Y, W)2 S 1 and this implies that |A,| S 1 for each i. Therefore, |/\1...Am| = 1 iff A? = w(e2,-_1,e2,-)2 = 1 for all i = 1, ..,m. This is equivalent to saying that e2,_1 = iJeg, for all i = 1, .., m in other words TpM is a complex space. For more details see [8]. Theorem 1.2 : (Federer’s Argument) Let X 2" be a thler manifold. Let : M 2"“ —> X be a compact complex sub- manifold. Then uol() g vol($) where 5 : Mm —> X is any real 2m-dimensional submanifold homologous to M rel boundary. (with equality if and only if Mm is also complex.) Proof: Let (I) : M —> X be a compact complex submanifold and 71>- : M —> X is any real 2m-dimensional submanifold homologous to M rel boundary. Take fl : W2”+1 —> X as the collection of simplices mapping to X such that M—M2BW. We will first show that / mm = f 52/". M if Since w is closed dw'" = 0. Also the pullback and exterior derivative commutes with each other. So we get Oszfi‘dwmszdfi‘wm = from (by Stoke’s theorem). ow :/ —IB¢wm:/Btwm_/‘_fitwm M—M M M Since the restrictions of B" to M and M are ‘ and 75‘, respectively, we get =/ <1>*wm—/$‘wm=0 M H Now, we can get the Federer’s Argument as follows: 1 —¢ vol(M) = dvolM = — ‘wm = —1 (I) w'" l l —_ M m. M m. M S / d'UOl'M (by Wirtinger’s Inequality) M = vol (M) (with equality iff M is also complex). 1.1.2 Calibrations In 1982, Harvey and Lawson extended the fact that complex submanifolds of a Kahler manifold are volume minimizing in their homology classes to the more gen- eral context of calibrated submanifolds. In their paper they introduced four new examples of calibrated geometries. The first is the special Lagrangian calibra- tion which is a real n form defined on a 2n dimensional manifold with holonomy contained in SU(n). The other three are associative, coassociative and Cayley calibrations which occur in specific dimensions. Most of the definitions used here can be found in [7]. Let X be a Riemannian manifold and (f) be a p-form on X. At each point x E X, the comass of 45,, is defined as follows: ”‘15”: = Sup{< quéz >: g is a unit simple p—vector at x} DEFINITION 1.5 : A smooth p-form (b on a Riemannian manifold X is called a calibration if i) (15 is comass 1. inwza (X, gb) is called a calibrated manifold. Let (t be a smooth p—form of comass 1 on X. We will denote the collection of oriented p-planes at x E X by G (p, TzX ) We can identify this set with the vector space of p—vectors at x. Then we can define Q(d)) as follows: g(¢) = { £2 EG(piTxX)| < ¢i€3 >2 1}' DEFINITION 1.6 : A p-dimensional submanifold S C X is called a Q(¢) subman- ifold if TqS E 9((15) for all q E 3. One can also define the calibration as follows: Note that these two definitions are equivalent. DEFINITION 1.7 : A calibration is a closed p-form (b on a Riemannian manifold X n such that it restricts to each tangent p-plane of X n to be less than or equal to the volume form of that p-plane. DEFINITION 1.8 : The submanifolds of X n for which the p-form (b restricts to be equal to the Riemmanian volume form are said to be calibrated by the form (15. We will use the term calibrated geometry for the ambient manifold X, the calibration <15, and the collection of submanifolds calibrated by (1'). Recall that in the previous section we showed that complex submanifolds of a Kahler manifold are volume minimizing in their homology classes, so if no denotes the Kahler form and if d)? = 91,—: then q)? is the calibration and the collection of complex submanifolds are the submanifolds calibrated by ¢p~ Next we will give an example: EXAMPLE 1.1 : Take w = dx in R”. We will find the calibrated geometries associated to w = dx. The comass of w 2 Sup {w(e) : |e| = l,e is a vector in R2} We can write 6 in terms of the basis: e 2 age; + b-giy such that |e| = W = 1. Then we get w(e) = a => Comass(w) 2 Sup {a1 |a| S 1}. Piom m = 1 we see that if comass: 1 then e = 8%. Therefore the associated 1-dimensional calibrated submanifolds of R2 will be straight lines parallel to the x-axis. In section 1.1.1 we showed that the complex submanifolds are volume mini- mizing in Kahler manifolds by proving Wirtinger’s Inequality and the Federer’s Argument. One can easily obtain similar properties for calibrated submanifolds: Let (X, gt) be a calibrated manifold. )If S IS a compact oriented p—dimensional submanifold of X, 1/S()’) B (with some singular fibres), such that X is obtained by finding some suitable compactification of the dual of this fibration. 13 1.3 Organization The thesis is organized as follows. In chapter 2 we review the deformation the- ory of special Lagrangian submanifolds in a Calabi-Yau manifold and explain R.C.McLean’s result. We will also extend this result to symplectic manifolds with non-integrable almost complex structure. CHAPTER 2 Deformations of Special Lagrangian Submanifolds In this chapter we will prove that the moduli space of all infinitesimal deformations of a smooth compact special Lagrangian submanifold L in a symplectic manifold X within the class of special Lagrangian submanifolds is a smooth manifold of dimension b1(L), the first Betti number of L. 2.1 Deformations in a Calabi-Yau Manifold In [10], McLean proved the following theorem which says that the moduli space of nearby submanifolds of a smooth compact special Lagrangian submanifold L in a Calabi-Yau manifold X is a smooth manifold and its dimension is equal to the dimension of ’Hl(L), the space of harmonic 1-forms on L. In what follows, X will denote a 2n-dimensional Calabi-Yau manifold with a Kahler 2-form w and 14 15 a nowhere vanishing holomorphic (n, 0)-form {zu + ifi, where u and O are real valued n—forms. Theorem 2.1 : The moduli space of all infinitesimal deformations of a smooth, compact, orientable special Lagrangian submanifold L in a Calabi-Yau manifold X within the class of special Lagrangian submanifolds is a smooth manifold of dimension equal to dim(7-ll(L)). REMARK 2.1 : R.C.McLean’s theorem is a tool to show the existence of non- explicit examples. It says that given one compact special Lagrangian submanifold L, there is a local finite dimensional moduli space of deformations whose dimension is equal to the first Betti number b1(L). Hence starting with a set of real points (special Lagrangian submanifold) in a suitable Calabi-Yau and deforming one can assert the existence of compact special Lagrangian submanifolds. Proof of Theorem: For a small normal vector field V we define the deforma- tion map as follows, F: P(N(L)) —> 92(L) ®Q"(L) 1’00 = ((expv)'(—w)i (expv)‘(1m(€)) The deformation map F is the restriction of —w and I m(£ ) to Ly and then pulled back to L via (expv)“. Here N (L) denotes the normal bundle of L, I‘(N(L)) the space of sections of the normal bundle, and {22(L), Q"( L) denote the differential 2-forms and n-forms, respectively. Also, expV represents the exponential map which gives a diffeomorphism of L onto its image LV in a neighborhood of 0. 16 Recall that the normal bundle N (L) of a special Lagrangian submanifold is isomorphic to the cotangent bundle T‘(L). Thus, we have a natural identification of normal vector fields to L with differential 1-forms on L. Furthermore, since L is compact we can identify these normal vector fields with nearby submanifolds. Under these identifications, it is then easy to see that the kernel of F will correspond to the special Lagrangian deformations. We compute the linearization of F at 0, dF(O) : P(N(L)) ——> 92(L) ®Q"(L) where dF(0)(V) = 52m»... = 5%[exp:v(—w),exp{v(fi)lli:o =[—(£vw)|L, (£vfi)|L] where .CV denotes the Lie derivative. Using the Cartan Formula, we get: = (-(ivdw + d(ivw))li. (ivdfi + d(ivfi))lL) = (—d X and since we work with closed forms exp}; and i“ give the same map in cohomology. Then [exp‘v(,6)] = [i*(fi)] = [fllL] = 0 and [exp{,(w)] = [i‘(w)] = [wIL] = 0 since L is special Lagrangian. So the forms in the image of F are cohomologous to zero and they are exact forms. Now, one can easily show that for any given exact 2—form a and exact n-form b we can solve for v that satisfies the equation dv = a and d * v = b. Hence dF(0)(V) is surjective and after completing the space of differential forms with appropriate norms and using the Banach space implicit function theorem and elliptic regularity we can conclude that F‘1(0, 0) is a smooth manifold with tangent space at 0 equal to 741(1), [10]. 2.2 A New Interpretation of Special Lagrangian Submanifolds In this section our aim is to improve R.C.McLean’s result which is explained in section 2.1 to symplectic manifolds. Precisely, we want to show that the moduli space of all infinitesimal special Lagrangian deformations of L in a symplectic manifold with non-integrable almost complex structure is also a smooth manifold 18 of dimension b1 (L). We will prove this by extending the parameter space of special Lagrangian deformations, in other words by using a modified definition of special Lagrangian submanifolds. First, we will explain why we need to change the classical definition of special Lagrangian submanifolds in terms of the calibrated form. Recall that McLean showed the surjectivity of the linearized operator in the Calabi-Yau case by a cohomology argument. This is possible because in the Calabi-Yau case the complex (n, O)-form 5 is closed but when we try to extend this result to symplectic manifolds with non-integrable almost complex structure (i.e 5 is no longer closed) we cannot use the same cohomology argument. So we have to seek some other ways to prove the surjectivity. One way is to change the deformation map slightly and that’s the main reason for us to use a modified definition of special Lagrangian submanifolds. Recall that a Lagrangian submanifold L of a Calabi-Yau manifold is special Lagrangian if I m(€ )| L E 0, where 5 is a nowhere vanishing, closed, complex (n, 0)- form. In our case, we will drop the assumption that 6 is closed (i.e d6 75 0) and introduce a new parameter 6 for the deformations. Then the condition Im(§) | L :— 0 will be replaced by I m(e‘0{ )| L E 0 in the definition of special Lagrangian subman- ifolds. In what follows, X will denote a 2n-dimensional symplectic manifold with sym- plectic 2-form w, an almost complex structure J which is tamed by w, the com- patible Riemannian metric g and a nowhere vanishing complex valued (n, 0)-form {=11 + ifl, where u and H are real valued n-forms. We say f is normalized if the following condition holds: 19 <—1)"<"-1>/2 92(L) $Q"(L) F(V,0) = ((expv)’(-w)i (expv)‘(Im(e“’€)) The deformation map F is the restriction of —w and I m(ef9£) to LV and then pulled back to L via (expv)‘ as in [13]. Here N (L) denotes the normal bundle of L, I‘(N(L)) the space of sections of the normal bundle, and 92(L), (N(L) denote the differential 2-forms and n-forms, respectively. Also, expV represents the exponential map which gives a diffeomorphism of L onto its image LV in a neighbourhood of O. 21 Recall that the normal bundle N (L) of a special Lagrangian submanifold is isomorphic to the cotangent bundle T‘(L). Thus, we have a natural identification of normal vector fields to L with differential 1-forms on L. Furthermore, since L is compact we can identify these normal vector fields with nearby submanifolds. Under these identifications, it is then easy to see that the kernel of F will correspond to the special Lagrangian deformations. We compute the linearization of F at (0,0), dF(0,0) : I‘(N(L))> 522(L) a; Q"(L) where dF(O, O)(V, 0) = gt-Fav, 30)|.:o,.=o + 333mm Santana Therefore, %F(tV, 30)|.:o,.=o + 333F(tV, 59)|t=0.s=0 = %[exp;V(—w), exp;,,(1m((ees(sa) + isin(s6))(u + mm |t=0,s=o +§§iexp:v(—w>,expi<1m< + isin(89))(u + ifi))llt=0.s=o =[—(va)|L, (CvullL ' Sin(39)|s=o +(5vfillL ° 008(39)|e=o + ((expiv It) ° C08(39) ' 9 - (expiv fl) 'Sin(89) ' 9)lt=o..=ol = [“(vallL, LVflIL ' COS(36)|320 + ((exp,’v M) ‘C05(59) ° 0)lt=0,r=o] Here CV represents the Lie derivative and one should notice that expz’v ultzo is just the restriction of u to L which is equal to 1 by our assumption that the initial 22 value of 6 is 0. Also, on a compact manifold L, top dimensional constant valued forms cor- respond to ’H"(L), the space of harmonic n-forms on L and there is a natural identification between the reals and harmonic n-forms. Therefore, 6 = 6de1 will play the role of a harmonic n-form in our calculations. Using the Cartan Formula, we get: = (—(ivdw + d(ivw))lLi (ivdfi + d(iv5))IL + 9) = (—d(ivw)|L, (ivdfi + defiUlL + 9) = (dv,C + d * v + 6), where C = iv(dfi)|L Here iv represents the interior derivative and v is the dual 1-form to the vector field V with respect to the induced metric. For the details of local calculations of d(ivw) and d(ivfi) see [10]. Hence dF(O, 0)(V, 6) = (dv,C + d * v + 6). Let x1, x2, ..., xn and x1, x2, ..., x2" be the local coordinates on L and X, respec- tively. Then for any given normal vector field V = (V1623+1 , ..., Vnfi) to L we can show that C = iv(d3)|L = —n(Vi '91 + + V" - gn)dvol where g,- (0 < i S n) are combinations of coefficient functions in the connection-one forms. One can also decompose the n-form C 2 da + d‘b + h2 by using Hodge Theory and because C is a top dimensional form on L, C will be closed and the equation becomes dF(0,0)(V,6) = (dv,da + d * v + h2 + 6) for some (n — 1)-form a and 23 harmonic n-form I12. The harmonic projection for C = —n(V1.g1 + + Vn.g,,)dvol is (/ —n(V1.g1 + + Vn.g,,)dvol)dvol and therefore one can Show that L do = —n(V1.g1 + + Vn.gn)dvol+(n/(Vl.g1 + + Vn.gn)dvol)dvol and L h; = (—n/(Ifi.gl + + Vn.gn)dvol)dvol. L REMARK 2.3 : One should note that the differential forms a and hg both de- pend on V and therefore should be explored carefully in order to understand the deformations of special Lagrangian submanifolds. After completing the space of differential forms with appropriate norms, we can consider F as a smooth map from Cl’°‘(fll(L)) xR to Co’“(fl2(L)) and Co'“(Q"(L)), where CW9) = {f 6 0km [Dinah < oo. lvlsk} and [flan = Sup dist§f§12.f(y)) in Q. d' t '1 :L',y€fl, x¢y ( 1‘ (21y)) The Implicit Function Theorem says that F_1(0, 0) is a manifold and its tangent space at (0,0) can be identified with the kernel of dF. (dv) $(C + d * v + 6) = (0,0) implies dv=0andC+d*v+6=da+d*v+h2+6=0. The space of harmonic n-forms ’H"(L), and the space of exact n—forms dfl"'1(L), on L are orthogonal vector spaces by Hodge Theory. Therefore, dv = 0 and da+d*v+h2+6 = 0 is equivalent to dv = 0 and d*v+da = 0 and h2+6 = 0. 24 One can see that the special Lagrangian deformations (the kernel of dF) can be identified with the 1-forms on L which satisfy the following equations: (i) do = 0 (ii) d * (’U + KM) = 0 (iii) h2 + 6 = 0. Here, 5(2)) is a linear functional that depends on v and h; is the harmonic part of C which also depends on v. These equations can be formulated in a slightly different way in terms of decompositions of v and *a. If v z: dp -l- d‘q + h] and *a = dm + d‘n + h3 then we have (i) dd*q = 0 (ii) A(p :l: m) = 0 (iii) h2 + 6 = o. This formulation of the solutions will help us to prove the surjectivity of the linearized operator without using n(v). REMARK 2.4 : When 6 = C, the infinitesimal deformations of 6 give no additional special Lagrangian deformations simply because there cannot be two different har- monic representatives in the same cohomology class. Therefore, one can obtain McLean’s result by fixing 6 = 0' along the deformations for some constant C and since dfi|L = 0 in the integrable case, do = 0 and h; = 0. Hence the deformations correspond to 1-forms which satisfy the equations dv = 0 and d * v + 6 = 0. Next, we need to show that the deformation theory of special Lagrangian sub- 25 manifolds is unobstructed. In order to use the implicit function theorem, we need to show that the linearized operator is surjective at (0, 0). Recall that the deformation map, F: I‘(N(L))> 92(L) @Q”(L) is defined as follows: F(V, 6) = ((expv)‘(-w)i (expv)‘(Im(€‘9€))- Even though I m(ewC ) is not closed on the ambient manifold X, the restriction of this differential form is a top dimensional form on L, and therefore it will be closed on L. On the other hand, no is the symplectic form which is by definition closed on X. Therefore, the image of the deformation map F lies in the closed 2-forms and closed n-forms. At this point we will investigate the surjectivity for w and I m(ewC ) separately. We have the following diagrams for dF = dF1 $dF2 with natural projection maps projl and projg : dFl :I‘(N(L)) —“'—> 02(L)p1ji dim) and, ng : I‘(N(L)) x R “(its)” (2"(L) ”1’? don-la.) gaunm We will show that the maps dFl and ng are onto dQl(L) and dfl"’1 (L) Q ’H“(L), respectively. Therefore, for any given exact 2-form x and closed n-form y = u + z in the 26 image of the deformation map (here u is the exact part and z is the harmonic part of y), we need to show that there exists a 1-form v and a constant 6 that satisfy the equations, (i) dv = x (ii) d * (v + K(v)) = u (iii)h2 + 6 = z. alternatively, we can solve the following equations for p, q and 6. (i) dd‘q = x (ii) A(p :l: m) = *u (Here, the star operator it is defined on L) (111) I12 + 9 = 2. For (i), since x is an exact 2-form we can write x = d(dr + d's+harmonic form) by Hodge Theory. Then one can solve (i) for q by setting q = s. For (ii), since Am 2 d‘dm = d‘ * a = *d at: *a = j: :1: da, A(p i m) = Ap :t Am 2 Ap :1: *da (here a depends on p) = Ap :t (—n(V1.g1 + + Vn.g,,) + (n/(Vlgl + + Vn.g,,)dvol )) = *u L Since V :— (V1, ..., V") is the dual vector field of the one form v :2 dp + d‘q + hl we can write the equation above as Ap :l: (—n(v - g) + (n/L(v - g)dvol )) = *u = Ap :t (—n(dp + d‘q + h1)-g + (n/(dp + d’q + h1)-gdvol )) = *u L where v - 9 represents the action of the one form v on the vector field 9 = (91, .., g") 27 and n/ (dp + d‘q + hl -g)dvol is the harmonic projection of —n(dp + d‘q + hl) ~g. L Then we get Ap :t n(—(dp - g) + fdp - gdvol )2 *u q: n[—(d"q + h1)-g + [(d‘q + h1)-gdvol]. L L For simplicity we put *u IF n[—(d"q + h) -g + [(d‘q + h) ~gdvol] = h. Since L / =1: u = 0 and /(d‘q + hl) - gdvol is the harmonic projection of (d‘q + h) -g, we L L get/h20. L Since L is a compact manifold without boundary, by Leibniz Rule, / dp - g dvol: — f p - divg dvol and the equation becomes L L Api n(—(dp - g) — [p - divg dvol) = h. L Then by adding and subtracting p from the equation I (A — Id)p = [in(-—(dp - g) — [p - divg dvol) — p + h] and L p = (A — Id)‘1[ ..... ]p + E = [C(p) + E, where h = (A — Id)“1h. and since ”(A — Id)‘1/p-divg||L§ _<_ CI/p-divgl S C||p||L2, [C(p) is a compact L L operator which takes bounded sets in L2 to bounded sets in L]. Also note that we assumed here 1 ¢ specA, and if this is not the case then we can modify the above argument by adding and subtracting Ap, A sf specA from the equation. Next we will show that the set of solutions of the equation Ap :l: n(—(dp - g) — f p - divg dvol) = 0 is constant functions and therefore of dimension 1. Note that L this set of solutions also satisfy the equation (Id — IC)(p) = 0 28 Note that / p - divg dvol is a constant which depends on p. We denote this as C (p) At maximum values of p, Ap will be negative which implies that C (p) S 0 and at minimum values of p, Ap will be positive which implies that C (p) 2 0 so C (p) should be zero. Then the maximum principle holds for the equation Ap i n(—(dp - g)) = 0 and since L is a compact manifold without boundary the solutions of this equation are constant functions. Hence the dimension of the kernel of (Id — IC) is one. Next we find the kernel of (Id — IC"). /L(Ap i n(-(dp - 9) - pr - divg))q(y)dy = [1913401) i n/ - (div - g)q(y)dy - 71/ (pr ' divgmyldy L L L = [LPG/Ma i n/L + (pdiv(g ~ q)(y)dy - n/LM (x) 'xdivfi Mfr. yldydx = /p(y)Aq :t n/ + (pdiv(g - q)(y)dy - n/pw) - divg(y) fr q(x)dxdy L L L = [LPG/)(Aq i n(+di'v(g - q) - divg fr. q(x)dmldy Since we assumed that 1 ¢ specA, dim ker(Id—IC*)(A—Id) = dim ker (Id—1C") and the kernel of (Id — IC“) is equivalent to the solution space of the equation Aq :t n(+div(g - q) — divg fL q(x)dx) = 0 By Fredholm Alternative, l 2] the dimension of this kernel is 1 and one can Check that a constant function q = 1 satisfies this equation, therefore the kernel consists of constant functions. Moreover these functions satisfy the compatibility condition fh.q = 0. 29 Then by Fredholm Alternative we can conclude the existence of solutions of the equation Ap i (—n(V1.g1 + + Vn.g,,) + (n/(V1.g1 + + Vn.g,,)dvol )) : *u L (iii) is straightforward. The only thing remaining is to show that the image of the deformation map F1 lies in dfll(L) and the image of F2 lies in dfln’l(L) ®’H"(L). For w, we can follow the same argument as in [13]. Since expV : L—>X is homotopic to the inclusion i : L—>X, exp;, and i“ induce the same map in coho- mology. Thus, [exp§, (w)] = [i‘(w)] = [wlL] = 0 . So the forms in the image of F is cohomologous to zero. This is equivalent to saying that they are exact forms. For I m(ei9 C ), we cannot follow the same process, because it is not a closed form on the ambient manifold X and therefore does not represent a cohomology class. But by our construction of our deformation map, it is obvious that the image lies in d0"‘1(L) Gianna). One can find the dimension of this manifold by comparing the operators d + *d*(v) and d + *d*(v + n(v)). Since C = iv(dfi)|L = —n(V1.g1 + + Vn.g,,)dvol it is easy to see that the extra term *d"(n(v)) contains no derivatives of v and this implies that the linearized operators d + *d"(v) and d + *d‘(v + n(v)) have the same leading term. Also it is known that the index of an elliptic operator is stable under lower order perturbations. Since the dimension of the kernel of d + *d‘ is b1(L_) + 1 and the dimension of its cokernel is 1 as a map from I‘(N(L))xR—> 30 dQI(L) $dfln’1(L)®’H"(L), we can conclude that both the index of d + *d“(v) and d + *d‘ (v + n(v)) are equal to b1(L). Hence the dimension of tangent space of special Lagrangian deformations in a symplectic manifold is also b1(L), the first Betti number of L. Therefore, dF is surjective at (0,0) and by infinite dimensional version of the implicit function theorem and elliptic regularity, the moduli space of all infinites- imal deformations of L within the class of special Lagrangian submanifolds is a smooth manifold and has dimension b1(L). [1] l2] [6] [7] [8] [9] [10] [11] BIBLIOGRAPHY Bryant, R.L. Some examples of special Lagrangian Tori, math.DG / 9902076 Gilbarg, D. and 'ITudinger S.N.Elliptic Partial Difi'erential Equations of Sec- ond Order, Springer-Verlag (New York 1977) Gross, M. Special Lagrangian Fibrations I: Topology, Integrable Systems and Algebraic Geometry (Kobe/ Kyoto 1997), 156-193 World Scientific. Gross, M. Special Lagrangian Fibrations II: Geometry, alg-geom/ 9809072 Harvey, F.R. Spinors and Calibrations, Perspectives in Mathematics, Vol 9, Academic Press, Inc. Hitchin, N. The moduli space of special Lagrangian submanifolds, dg- ga/9711002 Harvey, RR. and Lawson, H.B. Calibrated Geometries, Acta. 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