Special Lagrangian Geometry

碩士論文

Department of Mathematics College of Science

National Taiwan University Master Thesis

特殊拉格朗日球面存在性問題之探討

On the Existence Problem of Special Lagrangian Spheres

邱詩凱 Shih-Kai Chiu

指導教授：王金龍 博士 Advisor: Chin-Lung Wang, Ph.D.

中華民國 103 年 7 月 July 2014

在這趟為期兩年的數學旅程中，我最感謝的就是我的指導教授 –王金龍教授。

是王老師在我大四時開設的高等微積分課程，引領我進入數學這門我從未認真思 考過的科目。大學時的我讀的是工程，求學的過程中跌跌撞撞，幾番轉折才來到 了數學的大門。剛開始的時候，我在數學抽象、精緻的表象中迷失了，而艱澀的 課程加上幾乎等於零的基礎，讓我在研究所中逐漸喪失自信。是王老師讓我漸漸 明白，數學不僅僅是我所見到的表象，不僅僅是書上沒有生命的文字符號。是王 老師讓我明白，一個人應該以不卑不亢的態度來面對數學，並熱愛數學。這兩年 的時光，尤其是王老師的尊遵教誨，改變我人生之大，難以言表。

接著我要感謝我的兩位口試委員，張樹城教授與蔡忠潤教授。張老師開設的複 瑞奇流課程，讓我對當前的幾何分析有初步的了解。蔡老師的辛幾何課程讓我頭 一次對一門專門科目產生莫大的興趣，而每當我在論文上遇到難題時，蔡老師的 建議總是令人獲益良多。另外我要感謝齊震宇教授。我有幸在入學時，修到齊老 師開設的一系列課程，其嚴謹的內容和認真的教學態度，讓我明白身為一個數學 家應當具有的熱情與責任感。

我要感謝呂勇賢同學，在 Matlab 模擬均曲率流上給予的協助。我要感謝我的 大學同學李致遠，在論文格式上的協助。我要感謝李宗儒學長，他總是能夠迅速 理解我模糊的問題，給予我清晰而明確的指導。而不論在數學的學習與呈現，乃 至於與他人的互動，學長都讓人學習到甚多。

最後我要感謝我的家人和女朋友。我的父母對於我攻讀數學一事，乃至於求學 上的種種選擇，都給予我很大的自由。而我的女朋友匯詞，總是能在我鑽牛角尖，

陷入數學與情緒的拉扯時，給予我很大程度的支持與鼓勵。

感謝你們。

ii

在 Seidel 的博士論文 [Sei97] 中，他與他的指導教授 Donaldson 證明，若一緊緻 凱勒流形 (compact Kähler manifold) 擁有一個尋常退化 (ordinary degeneration)，則 此凱勒流形內存在拉格朗日球面 (Lagrangian sphere)。這個結果引發以下的延伸問 題：如果此凱勒流形為一卡拉比 -丘流形 (Calabi-Yau manifold)，我們是否能夠在 其中找出一個特殊拉格朗日球面 (special Lagrangian sphere)？透過文獻回顧，我們 將探討特殊拉格朗日子流形 (special Lagrangian submanifolds) 的基本知識，以及球 面的切叢 (the cotangent bundle of sphere) 上的瑞奇平坦度量 (Ricci-flat metrics)。在 論文的最後，我們透過均曲率流 (mean curvature flow) 來探討一維的情形。

iii

In his PhD thesis[Sei97], Paul Seidel and his advisor Simon K. Donaldson gave two proofs showing that a vanishing cycle in a Kähler manifold admitting an ordinary degener- ation can be chosen to be Lagrangian. This gives rise to the question whether the vanishing cycle is special Lagrangian if the manifold is Calabi-Yau. We investigate this problem by reviewing the geometric aspect of special Lagrangian manifolds and the Ricci-flat met- rics on the noncompact local model, namely the cotangent bundle of sphere. Finally, we approach this problem in dimension one through mean curvature flow.

iv

口試委員會審定書 i

謝辭 ii

中文摘要 iii

Abstract iv

1 Introduction 1

2 Special Lagrangian Geometry 2

2.1 Definitions and Basic Results . . . . 2 2.2 McLean’s Theorem . . . . 3 2.3 Geometric Structures on the Local Moduli Spaces . . . . 9

3 Ricci-flat metrics on T^∗Sⁿ 15

3.1 Existence of the Metric . . . . 15 3.2 Completeness of the Stenzel Metric . . . . 19 3.3 Special Lagrangian Structures . . . . 21

4 Existence of Lagrangian Spheres 23

4.1 Seidel’s Proof . . . . 24 4.2 Donaldson’s Proof . . . . 27

5 Discussion on the Main Problem 29

5.1 Formulation of the Main Problem . . . . 29 5.2 Results in n = 1 . . . . 30

v

Introduction

In his thesis[Sei97], Paul Seidel and his advisor Simon K. Donaldson showed that if a compact Kähler manifold admits an ordinary degeneration, then it contains a Lagrangian sphere. This motivates us to ask further that if the manifold is in fact Calabi-Yau, is it possible to find a special Lagrangian sphere in it? If this is the case, then by a theorem of McLean [McL96], this special Lagrangian sphere is rigid, provided that n≥ 2. This is a significant property in algebraic geometry.

We investigate this problem in the following way. In Chapter 2 we review the special Lagrangian geometry and McLean’s theorem on special Lagrangian deformations. The geometric structures on the local moduli space, due to [Hit97], is then reviewed. In Chap- ter 3 we follow Stenzel’s approach to contruct a Ricci-flat metric on the cotangent bundle of a sphere. In Chapter 4, we review the two methods to contruct a Lagrangian sphere in a compact Kähler manifold admitting an ordinary degeneration. Finally, in Chapter 5 we discuss possible approaches to the main problem we are concerned about, and describe some elemenetary results in the one dimensional case.

1

Special Lagrangian Geometry

2.1 Definitions and Basic Results

The main reference of the following contents on special Lagrangian geometry is [Joy03].

Let (M, g) be a Riemannian manifold. To each oriented k-plane V ⊂ TxM , we can assign a volume form vol_V, which is a k-form on V .

Definition 2.1.1. Let (M, g) be a Riemannian manifold and let ϕ be a closed k-form on M . We say that ϕ is a calibration on M if for every oriented k-plane V on M , we have ϕ|_V ≤ volV. Here ϕ|_V = α· volV for some α ∈ R and by ϕ|_V ≤ volV we mean α ≤ 1.

A k dimensional submanifold N of M is said to be calibrated by ϕ if for each x∈ N we have vol_TxN = ϕ|TxN.

Proposition 2.1.2. Let (M, g) be a Riemannian manifold, ϕ a calibration and N a sub-

manifold calibrated by ϕ. Then N is volume-minimizing among its homology class.

Definition 2.1.3. The tuple (Xⁿ, J, ω, Ω) is a Calabi-Yau manifold if (Xⁿ, J, ω) is a Käh- ler manifold of complex dimension n and Ω is a covariantly constant (n, 0)-form such that

ωⁿ

n! = (−1)^n(n2−1)(i

2)ⁿΩ∧ ¯Ω.

By standard linear algebra, it can be shown that if (X, J, ω, Ω) is a Calabi-Yau mani- fold, then the real part of Ω, Re Ω, is a calibration. From the calibrated geometry point of view, special Lagrangian submanifolds are the ones that is calibrated by Re Ω.

Definition 2.1.4. Let (X, J, ω, Ω) be a Calabi-Yau manifold. A submanifold calibrated by Re Ω is called a special Lagrangian submanifold, or SL n-fold for short.

2

A special Lagrangian submanifold is indeed a Lagrangian submanifold with respect to the Kähler form, with an additional feature:

Proposition 2.1.5. Let (Xⁿ, J, ω, Ω) be Calabi-Yau, and let L be a real n-dimensional submanifold of X. Then L is a special Lagrangian submanifold of X if and only if

Im Ω|L = 0, ω|L= 0.

The proof of this propostion can be found in section III.1 in the foundational paper [HL82].

Proposition 2.1.6. Let (X, J, ω) be a Kähler manifold and let L be a Lagrangian sub- manifold of L. Then

N L≃ T^∗L, where N L denote the normal bundle of L in X.

Proof. Since L is Lagrangian, J is an isomorphism between N L and T L. Now composing J with the musical isomorphism ♭ yields the desired isomorphism.

2.2 McLean’s Theorem

Theorem 2.2.1 ([McL96]). Let (X, J, ω, Ω) be a Calabi-Yau manifold, let L be a compact special Lagrangian submanifold of L, and let V ∈ C^∞(N L). Then V is a deformation vector field to a normal deformation through special Lagrangian submanifolds if and only if the 1-form (J V )^♭is harmonic.

We give a detailed proof here, following [Mar02]. In the following, we always write down the immersion f : L→ X explicitly for clarity. In fact, we only require f to be an immersion. The proof begins with a local observation.

Lemma 2.2.2. OnCⁿ, we have g₀ =∑n

j=1dx_j⊗ dxj+ dy_j⊗ dyj, ω₀ =∑n

j=1dx_j∧ dxj, J0 = i·, and Ω0 = dz1∧ . . . ∧ dzn, making (Cⁿ, J0, ω0, ω0, Ω0) a noncompact Calabi-Yau

manifold. Let V ⊂ Cⁿbe a special Lagrangian n-plane. Then for all ξ ∈ V^⊥, we have

(ι(ξ)ω0)|_V = (J0(ξ))^♭, (ι(ξ)Im Ω₀)|_V = ⋆(J₀(ξ))^♭,

where ⋆ is the Hodge star operator on V .

Proof. Since the equations are SU(n)-invariant, and since SU(n) acts transitively on the set of all SL n-planes inCⁿ, we may assume that V =Rⁿ, the real slice ofCⁿgenerated by _∂x^∂

1, . . . ,_∂x^∂

n. Then for k = 1, . . . , n,

(ι(_∂y^∂

k)ω₀)|_V = (−dxk)|_V

= (J_0∂y^∂

k)^♭0, and

(ι(_∂y^∂

k)Im Ω₀)|_V = (−1)^kdx₁∧ . . . ∧ ddx_k∧ . . . ∧ dxn

=− ⋆0dx_k

= ⋆₀(J_0∂y^∂

k)^♭0.

Corollary 2.2.2.1. Let (X, J, g, Ω) be a Calabi-Yau manifold and let f : L → X be a spcial Lagrangian submanifold. If ξ∈ C^∞(N L), then

f^∗(ι(ξ)ω) = (J (ξ))^♭, f^∗(ι(ξ)Im Ω0) = ⋆(J (ξ))^♭,

where ⋆ is the Hodge star operator on L.

Next we review the tubular neighborhood theorem, for the deformation occurs in a star-shaped tubular neighborhood of the initial submanifold.

Proposition 2.2.3 (Tubular neighborhood theorem). Let (M, g) be a Riemannian mani- fold, and let X be a submanifold of M , Then there exists an open neighborhood ˜U ⊂ NX containing the zero section such that

exp|U˜ : ˜U → M

is a diffeomorphism.

Shrinking ˜U if necessary, we may assume that ˜U ∩ Nx ⊂ Nxis star-shaped.

For the rest of this section, we always assume that (X, J, ω, Ω) is a Calabi-Yau mani- fold and that f : L → X is a compact special Lagrangian submanifold of X. Following the notations of the last proposition, we define

U = (J ˜U )^♭ ⊂ T^∗L,

U˜^∞ ={ξ ∈ C^∞(N L)| ξx ∈ ˜U ∀x ∈ L}, U^∞ ={η ∈ C^∞(T^∗L)| ηx ∈ U ∀x ∈ L}.

Let ξ ∈ C^∞(N L). Then there exists ϵ > 0 small enough such that tξ ∈ ˜U^∞for all

|t| < ϵ. This normal vector field ξ defines a deformation of X given by

f_tξ : L→ X, t ∈ (−ϵ, ϵ), f_tξ(x) = exp_{f (x)}tξ_x.

Note that f_tξ : L → X is a special Lagrangian submanifold if and only if f_tξ^∗ω = 0 and f_tξ^∗Im Ω = 0.

Lemma 2.2.4. Let η = (J ξ)^♭. Then

∂

∂t(f_tξ^∗ω)|_t=0= dη

and

∂

∂t(f_tξ^∗Im Ω)|_t=0 = d(⋆η).

Proof. Let ˜ξ be an extension of ξ on X.

∂

∂t(f_tξ^∗ω)|_t=0= f^∗(Lξ˜ω)

= f^∗(ι( ˜ξ)dω + dι( ˜ξ)ω)

= d(f^∗ι( ˜ξ)ω)

= dη,

where the last equality follows from the previous corollary. The other equation follows similarly.

From the above lemma it follows that, if ξ ∈ C^∞(N L) is a special Lagrangian defor- mation vector field, then (J ξ)^♭is necessarily a harmonic 1-form. Our goal is to show that this condition is unobstructed.

Fix k≥ 2. Define

U˜^k+1,a ={ξ ∈ C^k+1,a(N L)| ξx ∈ ˜U ∀x ∈ L}, U^k+1,a ={η ∈ C^k+1,a(T^∗L)| ηx ∈ U ∀x ∈ L},

and

F : ˜˜ U^k+1,a → C^k,a(∧ T^∗L) ξ 7→ − ⋆ fξ^∗Im Ω + f_ξ^∗ω.

Then we have ˜F (0) = 0, since f₀ = f : L→ X is a Special Lagrangian submanifold.

More generally, if ξ∈ ˜F⁻¹(0), then f_ξ : L→ X is a special Lagrangian submanifold. By composing with the isomorphism N L≃ T^∗L, we define F : U^k+1,a → C^k,a(∧

T^∗L) by

F ((J ξ)^♭) = − ⋆ fξ^∗Im Ω + f_ξ^∗ω.

The idea is to apply the implicit function theorem for Banach spaces to show that F⁻¹(0) is a manifold, parametrized by an open set of the finite dimensional vector space H¹of the harmonic 1-forms.

Theorem 2.2.5 (Implicit function theorem for Banach spaces). LetX1,X2 andY be Ba- nach spaces andU1 ⊂ X1,U2 ⊂ X2be open subsets both containing 0. Let

F :U1× U2 → Y (0, 0)7→ 0

be a map of class C^k such that the partial derivative F₂^′(0, 0) : X2 → Y is a topological linear isomorphism. Then there exists open setsW1 ⊂ X1andW2 ⊂ X2, both containing 0, and a unique C^kmapX : W1 → W2 such that

F⁻¹(0)∩ (W1× W2) ={(x1,X (x1))| x1 ∈ W1}.

By Hodge decomposition,

C^k+1,a(T^∗L) =H¹⊕ d^∗(C^k+2,a(∧2

T^∗L))⊕ d(C^k+2,a(L)). (2.1)

DefineX1 =H¹ andX2be the rest of the direct sum. LetU1 ⊂ X1 andU2 ⊂ X2 be open sets such that (0, 0)∈ U1× U2 ⊂ U^k+1,a. We restrict F toU1× U2.

Lemma 2.2.6. F : U1 × U2 → C^k,a(∧

T^∗L) has partial derivative F₂^′(0, 0) : X2 → C^k,a(∧

T^∗L) at (0, 0) in theX2-direction which acts as d^∗+ d.

Proof. Let η = (J ξ)^♭ ∈ X2. Then

F₂^′(0, 0)η = _∂t^∂(F (tη))|_t=0

= _∂t^∂(− ⋆ ftξ^∗Im Ω + f_tξ^∗ω)|_t=0

=− ⋆ (d ⋆ η) + dη

= (d^∗+ d)η.

Theorem 2.2.7 (Open mapping theorem). Let T : X → Y be a bounded linear map between Banach spaces. If T is surjective, then T is an open mapping. If T is bijective, then T is a toplinear isomorphism.

Theorem 2.2.8. F₂^′(0, 0) : X2 → C^k,a(∧

T^∗L) is a topological linear isomorphism onto the closed subspace

Y = d^∗(C^k+1,a(T^∗L))⊕ d(C^k+1,a(T^∗L)). (2.2)

Proof. We know that F₂^′(0, 0) = d^∗+ d. It’s clear that F₂^′(0, 0) mapsX2intoY injectively.

To see the reverse inclusion, let θ₁, θ₂ ∈ C^k+1,a(T^∗L). Then

d^∗θ₁ ∈ ∆C^k+3,a(L)

and

dθ₂ ∈ ∆C^k+3,a(∧2

T^∗L)

by Hodge decomposition. Therefore there exists f ∈ C^k+3,a such that d^∗(df ) = d^∗θ₁. simiarly, there exists ω ∈ C^k+3,a(∧₂

T^∗L) such that dd^∗ω + d^∗dω = dθ₂. Since d^∗dω is orthogonal to dθ₂, d^∗dω = 0. We conclude that

(d^∗+ d)(df + ω) = d^∗θ1+ dθ2,

where df +ω∈ C^k+2,a(T^∗L). Finally, by open mapping theorem, F₂^′(0, 0) is a topological linear isomorphism between Banach spacesX2andY.

To apply implicit function theorem, we show that image(F )⊂ Y.

Lemma 2.2.9. image(F )⊂ Y.

Proof. Note that there is a homotopy f_tξ between f and f_ξ. It follows that [f_ξ^∗Im Ω] =

[f^∗Im Ω] = 0 and [f_ξ^∗ω] = [f^∗ω] = 0. Consequently, ⋆f_ξ^∗Im Ω ∈ d^∗(C^k+1,α(T^∗L)) and f_ξ^∗ω ∈ d(C^k+1,α(T^∗L)).

It turns out that F : U1 × U2 → Y. By implicit function theorem, there exists open subests 0∈ W1 ⊂ U1 and 0∈ W2 ⊂ U2 and a unique mapX : W1 → W2such that

F⁻¹(0)∩ W1× W2 ={(η1,X (η1))| η1 ∈ W1}. (2.3)

This gives a smooth chart from the set of harmonic forms on L to the moduli space of special Lagrangian submanifolds near L. The dimension of this moduli space M = F⁻¹(0)∩(W1×W2), dimM , is equal to the first Betti number b¹(L) of L. This completes the proof of McLean’s theorem.

2.3 Geometric Structures on the Local Moduli Spaces

Let M be the local moduli space obtained in the last subsection. On M there is a natural Riemannian metric given by the L² inner product of harmonic 1-forms.

In [Hit97], Hitchin asked what is the natural geometrical structure on the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold.

By McLean’s theorem, we have an embedding M → H¹(L). But the construction of this embedding uses implicit funcition theorem, thereby not canoncial. Following [Hit97], we first show that there is a canoncial one.

Redefine f : M → X as the full local family of special Lagrangian deformations of L, whereM ≃ L × M. Let p : M → M be the projection map. For each t ∈ M, Lt = p⁻¹(t).

Let t₁, . . . , t_mbe alocal coordinate system of M . Of course, m = b¹(L) = dimH¹(L,R).

The tangent vectors _∂t^∂

j

on L_tdefine harmonic forms

θ_j = (ι(_∂t^∂

j)f^∗ω)|

Lt

.

These 1-forms can also be obtained as follows. Since f^∗ω = 0 on the fibre L_t(L_tis Lagrangian), there are 1-forms ˜θ_j onM such that f^∗ω = ∑

dt_j ∧ ˜θj. Then θ_j = ˜θ_j|_Lt, which coincides the previous definition.

Choose a basis A1, . . . Am ∈ H1(L,Z) modulo torsion. Let α1, . . . αm ∈ H¹(L,R) be the dual basis of A1, . . . A_m. Then we obtain a period matrix

λ_ij = ˆ

Ai

θ_j.

The matrix (λ_ij) is invertible, since θ_j are linearly independent.

Proposition 2.3.1. The 1-forms ξi =∑

λ_ijdt_j on M are closed.

Proof. Choose smoothly in each fibre L_ta circle representing A_i. Then we have a fibration

S^{1 //}Mi p

M

Define a 1-form ξ on M by

ξ = p_∗f^∗ω.

The push-down map p_∗ is the integration over the fibre. Since p_∗ sends closed forms to closed forms, ξ is closed. Now,

p_∗f^∗ω = p_∗(∑

dtj ∧ ˜θj)

=∑

dt_j ˆ

Ai

θ_j

= ξ_i.

Since the 1-forms ξ_i are closed, locally we can find functions u_i : M → R, well-

defined up to a constant, such that du_i = ξ_i = ∑

λ_ijdt_j. Since (λ_ij) is invertible, u₁, . . . , u_mdefine a coordinate chart. More invariantly,

Proposition 2.3.2. We have a coordinate chart u : M → H¹(L,R) defined by

u(t) =∑

u_i(t)α_i.

Moreover, this embedding is independent of the choice of basis, and is unique up to a translation.

Proof. Let the prime version denote another choice of basis. There exists an invertible matrix (Tij) such that Ai = TijA^′_j and αi = Tjiα^′_j. The period matrix transforms as λ_ij = T_ikλ^′_kj. Now,

∑u_iα_i =∑ ˆ ^t

du_i∧ αi

=∑ ˆ ^t

λ_ijdt_j ∧ αi

=∑ ˆ ^t

T_ikλ^′_kjdt_j ∧ αi

=∑ ˆ ^t

du^′_k∧ α^′k

=∑

u^′_kα_k^′.

Paralleling the procedure above, we have a similar result for Im Ω. Since Ltis special Lagrangian, f^∗Im Ω = 0 on the fibre L_t. Therefore there exist (n− 1)-forms ˜ϕj onM such that f^∗Im Ω = ∑

dt_j ∧ ˜ϕj. The restriction ϕ_j = ϕ_j|_Lt is independent of the choice of ˜ϕ_j. In fact, ϕ_j = (ι(_∂t^∂

j)f^∗Im Ω)|

Lt

and ϕ_j = ⋆θ_j by Corollary 2.2.2.1.

Let B₁, . . . , B_m ∈ Hn−1(L,Z) be the Poincaré duals of α1, . . . , α_m. We form a period matrix

µ_ij = ˆ

Bi

ϕ_j.

Then there exist coordinates functions v₁, . . . , v_m such that dv_i = µ_ijdt_j and an em- bedding

v : M → Hⁿ⁻¹(L,R)

given by

v(t) =∑ viβi,

where β₁, . . . , β_m ∈ Hⁿ⁻¹(L,R) are the dual basis of B1, . . . , B_m. The matrices (λ_ij) and (µ_ij) are related as follows:

Lemma 2.3.3.

∑

i

λ_ijµ_ik =∑

i

λ_ikµ_ij.

Proof.

ˆ

L

θ_j∧ ϕk= ˆ

L

θ_j∧ ⋆θk

= ˆ

L

θ_k∧ ⋆θj

= ˆ

L

θk∧ ϕj.

Using θ_j =∑

λ_ijα_i, ϕ_k=∑

µ_lkβ_land the fact that α_i and β_kare Poincaré duals, we get the identity.

u and v together give an embedding

F : M → H¹(L,R) × Hⁿ⁻¹(L,R)

by

F (t) = (u(t), v(t)).

The main result of [Hit97] is to show that the geometry of M is inherited from H¹(L,R)×

Hⁿ⁻¹(L,R). We first recall the linear geometry on V ⊕V^∗where V is a finite dimensional vector space. There is naturally a symplectic structure on V ⊕ V^∗given by

W ((v, α), (v^′, α^′)) = α^′(v)− α(v^′).

There is also a symmetric bilinear form G on V ⊕ V^∗given by

G((v, α), (v^′, α^′)) = 1

2(α^′(v) + α(v^′)).

It follows that manifold H¹(L,R) × Hⁿ⁻¹(L,R) has these two structures on the tangent space H¹(L,R) ⊕ Hⁿ⁻¹(L,R); in other words, H¹(L,R) × Hⁿ⁻¹(L,R) is a symplectic manifold together with a symplectic form W and an indefinite metric G. This indefinite metric restricted M is the L² metric:

Proposition 2.3.4. The L²metric g on M is F^∗G.

Proof. We have dF (_∂t^∂

j) = (∑

λijαi,∑

µlkβl). Thus for two tangent vectors,

F^∗G(a_j∂t^∂

j, b_k∂t^∂

k) = 1

2(a_jb_kλ_ijµ_ik+ a_kb_jλ_ikµ_ij) = a_jb_kλ_ijµ_ik,

where we sum over repeated indices. On the other hand,

g(a_j∂t^∂

j, b_k∂t^∂

k) = a_jb_k ˆ

L

θ_j∧ ⋆θk = a_jb_k ˆ

L

θ_j ∧ ϕk.

(Again we are summing over repeated indices.) Plugging θ_j =∑

λ_ijα_iand ϕ_k =∑ λ_ikβ_i into the above equation gives the equality.

Theorem 2.3.5. The map F embeds M in H¹(L)× Hⁿ⁻¹(L) as a Lagrangian submani- fold.

Proof. In local coordinates (u, v), the symplectic form W reads

W =∑

dui ∧ dvi.

Pulling back to M , we get

F^∗W = F^∗∑

dui∧ dvi

=∑

∂ ui

∂t_j

∂ vi

∂t_kdt_j∧ dtk

=∑

λ_ijµ_ikdt_j ∧ dtk.

But∑

iλ_ijµ_ik =∑

iλ_ikµ_ij. So W pulls back to 0.

Any Lagrangian submanifold of V × V^∗ ≃ T^∗V can be written locally as the image of the exact differential dϕ : V → T^∗V , where ϕ is a (locally defined) function. In our case, H¹(L)× Hⁿ⁻¹(L)≃ T^∗H¹(L), and the fibre coordinates v_j can be written as

v_j = dϕ(u_j) = _∂u^{∂ ϕ}

j.

Exchanging the roles of H¹(L) and Hⁿ⁻¹(L), we can also write uj = _∂v^{∂ ψ}

j for some locally defined function ψ on Hⁿ⁻¹(L). In these coordinates, the metric G can be written as

G = ∑

du_idv_i =∑ ∂²ϕ

∂u_i∂u_jdu_idu_j =∑ ∂²ψ

∂v_i∂v_jdv_idv_j.

As in the case of deformations of complex manifolds, one can ask when is M itself a special Lagrangian submanifold. The manifold V × V^∗has two constant m-forms, which are the generators of∧^mV^∗ and∧^mV . Following the previous notion, we can define a submanifold of V×V^∗as a special Lagrangian if it is Lagrangian and a linear combination of these constant m-forms vanishes. For further discussion we refer to [Hit97].

Ricci-flat metrics on T ^∗ S ⁿ

In this section, we introduce a local model of special Lagrangian spheres, namely the cotangent bundle of the standard n-sphere, T^∗Sⁿ. We equip T^∗Sⁿ with a Ricci-flat metric, called the Stenzel metric, and show that the zero section Sⁿis a special Lagrangian submanifold.

3.1 Existence of the Metric

First we fix coordinates by T^∗Sⁿ ={(x, ξ) ∈ Rⁿ⁺¹× Rⁿ⁺¹| |x| = 1, x · ξ = 0}. The group SO(n + 1,R) acts transitively on T^∗Sⁿby g· (x, ξ) = (gx, gξ), g ∈ SO(n + 1, R).

The general orbit of this action is the sphere bundle SO(n + 1)/SO(n− 1) ≃ Sⁿ× Sⁿ⁻¹, while the exceptional orbit being the zero section Sⁿ. It is well-known that there exists a diffeomorphism from T^∗Sⁿto the affine quadric

Qⁿ ={z = (z1+ . . . + z_n+1)∈ Cⁿ⁺¹|

∑n+1 j=1

z_j² = 1},

given by

(x, ξ)7→ z = x cosh(|ξ|) + isinh(|ξ|)

|ξ| ξ.

This diffeomorphism is equivariant with respect to the action of SO(n+1,R) on T^∗Sⁿ and the action of SO(n + 1,C) on Qⁿ. Thus we can view T^∗Sⁿas the complexification of the homogeneous space Sⁿ. We make T^∗Sⁿ into a complex manifold simply by pulling back the complex structure of Qⁿ. Our next goal is to describe a method to construct a

15

family of Ricci-flat metrics on Qⁿ.

Let τ be the restriction of the function∥z∥² =∑n+1

j=1z_j² to Qⁿ. By a simple calcu- lation we see that τ : Qⁿ → [1, ∞) is a strictly plurisubharmonic exhaustion on Qⁿ. We seek [Ste93] Ricci-flat Kähler potentials of the form ρ = f ◦ τ, where f : [1, ∞) → R is a smooth function. For later calculations, we fix a local frame on Qⁿas follows:





















v₁ = −zn+1 ∂

∂z₁ + z_1∂z^∂

n+1, v₂ = −zn+1 ∂

∂z₂ + z_2∂z^∂

n+1, ...

v_n = −zn+1 ∂

∂z_n + z_n∂z^∂

n+1

.

Let u¹, . . . , uⁿbe the dual frame. The Kähler form of ρ is given by

ω = i∂ ¯∂ρ

= i(f^′′∂τ ∧ ¯∂τ + f^′∂ ¯∂τ )

= i(f^′′τ_jτ¯k+ f^′τ_j¯k)u^j∧ u^k¯,

where τj, τ¯kdenote the differentiations in the directions of vj and ¯vk, respectively:

τ_j =−zn+1z¯_j+ z_jz¯_n+1, τ¯k=−¯zn+1z_k+ ¯z_kz_n+1, τ_j¯k=|zn+1|²δjk+ zjz¯k.

The Ricci-form of ρ is given by

Ric(ρ) = −i∂ ¯∂log det ρj¯k

=−i∂ ¯∂log det(f^′′τ_jτ¯k+ f^′τ_j¯k).

We can investigate further: