f -minimal Lagrangian Submanifolds in Kähler Manifolds with Real Holomorphy Potentials
aa r X i v : . [ m a t h . DG ] J a n f -minimal Lagrangian Submanifolds in K¨ahler Manifolds withReal Holomorphy Potentials WEI-BO SU
Abstract
The aim of this paper is to study variational properties for f -minimal Lagrangian sub-manifolds in K¨ahler manifolds with real holomorphy potentials. Examples of submanifolds ofthis kind incuding soliton solutions for Lagrangian mean curvature flow (LMCF). We derivesecond variation formula for f -minimal Lagrangians as a generalization of Chen and Oh’sformula for minimal Lagrangians. As a corollary, we obtain stability of expanding and trans-lating solitons for LMCF. We also define calibrated submanifolds with respect to f -volumein gradient steady K¨ahler–Ricci solitons as generalizations of special Lagrangians and trans-lating solitons for LMCF, and show that these submanifolds are necessarily noncompact.As a special case, we study the exact deformation vector fields on Lagrangian translators.Finally we discuss some generalizations and related problems. The stability properties of minimal Lagrangian submanifolds in K¨ahler manifolds is studiedby Chen [6] and Oh [27]. In particular, they derive a beautiful second variational formula asfollows. Let (
X, J, ω ) be a K¨ahler manifold with metric g , F : L X be a minimal Lagrangiansubmanifold, and let { F t } be a smooth family of compactly supported normal deformations of F with F = F and ddt (cid:12)(cid:12) t =0 F t = ξ ∈ Γ c ( N L ). Since L is Lagrangian, ξ is naturally identified witha 1-form α ξ := F ∗ ι ξ ω ∈ Γ( T ∗ L ). Then( δ V ) F ( L ) ( ξ ) := d dt (cid:12)(cid:12) t =0 V ( F t ) = Z L (cid:0) | dα ξ | g + | d ∗ g α ξ | − Ric( ξ, ξ ) (cid:1) dV g , (1)where g is the induced metric on L , and Ric is the Ricci curvature of g . A minimal La-grangian F : L X is called stable if ( δ V ) F ( L ) ( ξ ) ≥ ξ ∈ Γ c ( N L ), Lagrangianstable if ( δ V ) F ( L ) ( ξ ) ≥ ξ ∈ Γ c ( N L ) with α ξ being closed, and Hamiltonian stable if( δ V ) F ( L ) ( ξ ) ≥ ξ ∈ Γ c ( N L ) with α ξ being exact. From (1) one can deduce that (i) if Ric <
0, then any minimal Lagrangian in X is strictly stable, (ii) if ( X, J, ω ) is positive K¨ahler-Einstein, that is, Ric = c · g for some c >
0, then a minimalLagrangian in X is Hamiltonian stable if and only if λ (∆ g ) ≥ c , and (iii) if ( X, J, ω,
Ω) is a Calabi–Yau manifold equipped with Ricci-flat K¨ahler metric, then anyminimal Lagrangian submanifold in X is stable, and Jacobi fields on L are given bysolutions of harmonic 1-form equation dα = d ∗ g α = 0 , α ∈ Γ( T ∗ L ) . X, J, ω,
Ω) is special Lagrangian ,that is, F : L X is calibrated by Re( e − iθ Ω), or equivalently, F ∗ ω = F ∗ Im( e − iθ Ω) = 0 (2)on L for some phase θ ∈ S . This guarantees that if L is compact, F : L X is not onlyminimal but volume-minimizing in its homology class. McLean [23] shows that the linearizationof equation (2) is exactly the harmonic 1-form equation, thus any Jacobi field on L also inducesan infinitesimal special Lagrangian deformation. Moreover, the special Lagrangian deformationsare unobstructed and the moduli space of special Lagrangians is a smooth manifold of dimension b ( L ). f -stability of f -minimal Lagrangians and LMCF Solitons In this paper, we aim to generalize the above story to Lagrangian submanifolds which are“minimal” with respect to certain weighted volume functionals. Consider a smooth function f : X → R and the corresponding weighted volume form e − f dV g on a K¨ahler manifold ( X, J, ω ).An analogue quantity for Ricci curvature Ric on the metric measure space (
X, g, e − f dV g ) is thesymmetric 2-tensor Ric f = Ric + Hess f , called the Bakry–´Emery Ricci tensor. Since g is K¨ahler,Ric is Hermitian with respect to J . Hence it is natural to require Hess f to be Hermitian. This isequivalent to requiring that f to be a real holomorphy potential , that is, ∇ , f is a holomorphicvector field on X .Define the f -volume functional on the space of p -submanifolds by V f : ( F : L X ) Z L e − p m F ∗ f dV g ∈ R + ∪ { + ∞} . Notice that e − p m F ∗ f dV g is the volume of the induced conformal metric F ∗ ( e − fm g ) on L . A p -submanifold F : L X is a critical point with respect to V f if and only if the generalizedmean curvature vector H + p m ( ∇ f ) ⊥ = 0, where H is the mean curvature vector of F : L X .Such a submanifold is called an f -minimal submanifold.Under the above settings, we prove the following second variational formula for f -minimalLagrangian submanifolds: Theorem 1.1.
Assume that ( X, J, ω, f ) is a K¨ahler manifold with a real holomorphy potential,and F : L X is an f -minimal Lagrangian submanifold. Then for any compactly supportednormal variational vector field ξ on L , ( δ V f ) F ( L ) ( ξ ) = Z L { | dα ξ | g + | d ∗ f α ξ | − Ric f ( ξ, ξ ) } e − F ∗ f dV g , (3) where α ξ := F ∗ ι ξ ω is the -form on L associated to ξ , and d ∗ f is the adjoint of d in the weightedspace L (Λ ∗ T ∗ L, e − F ∗ f dV g ) . We call an f -minimal Lagrangian f -stable if ( δ V f ) F ( L ) ( ξ ) ≥ ξ on L , and the notions Lagrangian f -stable and Hamiltonian f -stable are defined analogously.Typical examples of K¨ahler manifolds admitting real holomorphy potentials are gradientK¨ahler–Ricci solitons (KR soliton for short), that is, K¨ahler manifolds satisfyingRic + Hess f = c · g. f is constant, this equation reduces to K¨ahler-Einstein equation. In this sense, gradientKR solitons are generalizations of K¨ahler-Einstein manifolds. For c = 0, the soliton is called steady , for c >
0, it is called shrinking , and for c <
0, it is called expanding . We then obtaina corollary of Theorem 1 . f -minimal Lagrangian in a steady or expanding gradient KR soliton is f -stable, and that a f -minimal Lagrangian in a shrinking gradient KR soliton is Hamiltonian f -stable if and only if λ (∆ f ) ≥ c , where ∆ f is the Witten Laplacian on L associated to g and F ∗ f .Our formula (3) can be applied to study the stability of soliton solutions for Lagrangian meancurvature flow (LMCF). In fact, if X = C m with standard Euclidean metric g and f ( z ) := ± | z | ,then ( C m , i, g , f ) is a shrinking/expanding gradient KR soliton and the f -minimal Lagrangiansare shrinking/expanding solitons for mean curvature flow , respectively, and if f ( z ) := h z, T i forsome fixed T ∈ C m , ( C m , i, g , f ) becomes a gradient steady KR soliton and the f -minimalLagrangians are translating solitons (see Example 1 and 2). By Theorem 1 .
1, we see that:
Corollary 1.2.
Every expanding soliton and translating soliton for Lagrangian mean curvatureflow is f -stable. The stability of soliton solutions to mean curvature flow under certain weighted volume func-tional was first studied by Colding–Minicozzi [7] for shrinking solitons in hypersurface case, andgeneralized to higher codimensional case by Andrews–Li–Wei [1], Arezzo–Sun [2], and Lee–Lue[16], but note that their functional (“entropy”) is different from the f -volume functional. Forstability of translating solitons, the second variation formula for translating hypersurfaces under f -volume was obtained by Xin in [35], Shahriyari [28] studied the stability of graphical translat-ing surfaces in R , and Yang [38] and Sun [30] studied the Lagrangian translating solitons. Inparticular, Yang [38] proved that every Lagrangian translating soliton is Hamiltonian f -stable,and Sun [30] showed that they are actually Lagrangian f -stable. f -special Lagrangians and Translating Solitons Next, we will focus on the case c = 0, so we let ( X, J, ω, f ) be a gradient steady KR soliton. In[4], Bryant shows that there is a holomorphic volume form Ω f such that ( X, J, ω, Ω f ) becomesan almost Calabi–Yau m -fold. The m -form Re Ω f is a calibration on X with respect to theconformal metric e − fm g . A key observation is that, this fact can be rephrase as that Re Ω f is acalibration with respect to the f -volume e − f dV g on X , and hence any submanifold calibratedby Re Ω f minimize the f -volume. We call such submanifolds f -special Lagrangians ( f -SLags) and view them not only as generalizations of special Lagrangians in Calabi–Yau manifolds, butalso generalizations of Lagrangian translating solitons in C m , since they evolved under LMCFby “translation” along the negative gradient vector field − ∇ f .It turns out that every f -minimal Lagrangians in a gradient steady KR soliton ( X, J, ω, f )can be viewed as an f -SLag, and hence the f -stability also follows from this point of view. The f -SLag deformations can be characterized by the solutions of the f -harmonic 1-form equation dα = d ∗ f α = 0 , α ∈ Γ c ( T ∗ L ) . Thus if L is compact, the moduli space of f -SLags is a smooth manifold with dimension b ( L ).But unfortunately we have a nonexistence result: Proposition 1.3.
There is no compact f -minimal Lagrangian in a gradient expanding or steadyKR soliton ( X, J, ω, f ) . f -SLags, one needs to impose suitable asymptoticconditions. As an experiment, we study the case when F : L C m is an exact Lagrangiantranslating soliton and assuming that the deformation is exact with weighted L potential. Weshow that such deformation must be trivial on L , that is, there is no nonzero weighted L f -harmonic function on L . Proposition 1.4.
Suppose F : L C m is a Lagrangian translating soliton and u is a functionon L with k u k L ( e − F ∗ f dV g ) < ∞ and ∆ f u = 0 . Then u ≡ . To study the deformation theory of Lagrangian translating solitons further, one needs toimpose more complicated asymptotic conditions and study the Fredholm theory of ∆ f in thecorresponding weighted spaces. On the other hand, the properties of f -harmonic functions onnoncompact f -minimal submanifolds might be useful for describing the topology at infinity. See[11], [12] for some results in this direction in hypersurface case.This paper is organized as follows. In Section 2 we introduce the K¨ahler manifolds withreal holomorphy potentials and f -minimal Lagrangian submanifolds. In Section 3 we provethe second variation formula for the f -volume and stability of solitons for LMCF. We study f -calibrated submanifolds and prove a noncompact result in Section 4. In the final section,some generalizations and related problems are discussed. Acknowledgements
Part of this paper were done when the author was visiting the Mathematics Institute in Univer-sity of Oxford as a Recognised Student supervised by Professor Dominic Joyce, from January2017 to June 2017. The author wants to express his gratitude to the Institute for the hospitalityand to Professor Joyce for his kind advices and suggestions. He would also like to thank hisadvisor Professor Yng-Ing Lee for her constant encouragements and supports, and ProfessorJason Lotay for the useful comments. The author is supported by MOST project 105-2115-M-002-004-MY3, and his visit to Oxford was also supported by National Center for TheoreticalSciences and Professor Dominic Joyce.
In the following, (
X, J ) will be a smooth, connected, complex manifold with dim R X = 2 m , and ω will be a K¨ahler form with K¨ahler metric g . The Levi-Civita connection of g will be denotedby ∇ , and the corresponding quantities with respect to ∇ , such as Hessian and curvature, willbe denoted by notations with overline.We will assume that there exists a function f : X → R such thatHess f ( J X, J Y ) = Hess f ( X, Y ) , (4)where Hess is the Hessian of f with respect to g . In fact, it is not hard to see that the followingconditions are equivalent. Proposition 2.1.
Let f : X → R be a smooth function. The following are equivalent:(i) ∇ , f is a holomorphic vector field,(ii) Hess f ( J X, J Y ) = Hess f ( X, Y ) for any X, Y ∈ Γ( T X ) , iii) J ∇ f is a Killing vector field. Such a function f is called a real holomorphy potential on ( X, J, ω ). Some properties ofK¨ahler manifolds admitting real holomorphy potentials can be found in Munteanu–Wang [24],[25]. Typical examples of manifolds of this kind are gradient K¨ahler–Ricci solitons : Example 1 (Gradient K¨ahler–Ricci Solitons) . Consider K¨ahler manifold (
X, J, ω ) togetherwith a smooth function f : X → R satisfyingRic + Hess f = c · g, (5)for some c ∈ R , where Ric is the Ricci curvature of g . Since by K¨ahler condition we haveRic( J X, J Y ) = Ric(
X, Y ), so f must satisfyHess f ( J X, J Y ) = Hess f ( X, Y ) . Thus f is a real holomorphy potential. The quadruple ( X, J, ω, f ) is called a gradient K¨ahler–Ricci solitons ( KR solitons for short). The gradient vector field ∇ f generates soliton solutionto K¨ahler–Ricci flow (KRF) ∂ t ω ( t ) = − ρ ( ω ( t )), where ρ ( ω ( t )) is the Ricci form of ω ( t ), in thefollowing way. Define g ( t ) := σ ( t ) ϕ ∗ t g, where σ ( t ) = 1 − ct and ϕ t is the flow on X generated by σ ( t ) ∇ f . Then it is straightforwardto verify that ω ( t ) satisfies the KRF with ω (0) = ω as long as σ ( t ) > c ∈ R : (i) When c <
0, (
X, J, ω, f ) is called a gradient expanding soliton. The KRF g ( t ) evolves themetric g by homothetically expanding the length scale since σ ( t ) ′ = − c >
0. For example,take (
X, J ) = ( C m , i ) with Euclidean metric ω , and f ( z ) := − | z | , then c = −
1. Theresulting expanding soliton ( C m , i, ω , f ) is called the expanding Gaussian soliton. (ii) When c >
0, (
X, J, ω, f ) is called a gradient shrinking soliton. The KRF g ( t ) evolves themetric g by homothetically shrinking the length scale since σ ( t ) ′ = − c <
0. For example,take (
X, J ) = ( C m , i ) with Euclidean metric ω , and f ( z ) := | z | , then c = 1. The resultingshrinking soliton ( C m , i, ω , f ) is called the shrinking Gaussian soliton. (iii) When c = 0, ( X, J, ω, f ) is called a gradient steady soliton. The KRF g ( t ) evolves themetric g by holomorphic reparametrizations ϕ t . For example, take ( X, J ) = ( C m , i ) withEuclidean metric ω , and f ( z ) := h z, T i with some fixed vector T ∈ C m , then ( C m , i, ω , f )is a steady soliton structure on C m .Note that if f is constant, condition (5) reduces to K¨ahler-Einstein condition. Hence gradientKR solitons can be viewed as generalizations of K¨ahler-Einstein manifolds. f -minimal Lagrangian Submanifolds Given a K¨ahler manifold with holomorphy potential (
X, J, ω, f ). Let F : L X be an immersed,oriented, connected submanifold. The induced metric will be denoted by g := F ∗ g , and the5orresponding quantities, such as Levi-Civita connection and curvature of g , will be denoted bynotations without overline. Define the f -volume functional on the space of p -submanifolds by( F : L X ) V f ( F ( L )) := Z L e − p m F ∗ f dV g . Let { F t : L X } t ∈ ( − ǫ,ǫ ) be a compactly supported normal variational of F : L X , with F = F and ddt (cid:12)(cid:12) t =0 F t = ξ ∈ Γ c ( N L ), then the first variation formula of the f -volume is given by ddt V f ( F t ( L )) (cid:12)(cid:12) t =0 = − Z L h H + p m ∇ f , ξ i e − p m F ∗ f dV g , (6)where H is the mean curvature vector of F : L X , and ( · ) ⊥ is projection to the normalbundle of L . Definition 2.2. A p -submanifold F : L X is called f -minimal if H + p m ( ∇ f ) ⊥ = 0 . If f is constant, f -minimal submanifolds are minimal submanifolds. Hence f -minimal sub-manifolds can be viewed as generalizations of minimal submanifolds.In the following, we will consider only Lagrangian submanifolds . Recall that an m -submanifold F : L m X in a symplectic manifold ( X m , ω ) is called Lagrangian if F ∗ ω = 0. If F : L X is Lagrangian, any compatible almost complex structure J on X gives rise to an isomorphismbetween normal bundle N L and tangent bundle
T L of L . Then by composing with the inducedmetric g we obtain an isomorphism between N L and T ∗ L . We will use the same notation as in[27] to denote such isomorphism. Definition 2.3.
Let F : L X be a Lagrangian submanifold. Define an isomorphism e ω : N L → T ∗ L by e ω ( ξ ) := F ∗ ( ι ξ ω ) , ξ ∈ Γ( N L ) . (7)In a K¨ahler manifold, Dazord [8] shows that the mean curvature vector H of a Lagrangiansubmanifold F : L X satisfies d e ω ( H ) = F ∗ ρ , where ρ = Ric( J · , · ) is the Ricci form. Thusif ( X, J, ω ) is K¨ahler-Einstein, then e ω ( H ) is closed, so it induces an infinitesimal Lagrangiandeformation. Furthermore, Smoczyk [29] shows that the Lagrangian condition is preserved bythe mean curvature flow ddt F t = H ( F t ) whenever ( X, J, ω ) is K¨ahler-Einstein. Therefore, it isreasonable to consider
Lagrangian mean curvature flow (LMCF) in K¨ahler-Einstein manifoldsand soliton solutions for LMCF.The next example explains the meaning of LMCF solitons and shows that they can be viewedas f -minimal Lagrangian submanifolds in C m . Example 2 (Soliton Solutions for LMCF) . Consider (
X, J ) = ( C m , i ), ω = Euclidean metric ω . (i) Define f ( z ) := | z | , then f is a real holomorphy potential and ∇ f ( z ) = z . Then any f -minimal Lagrangian submanifold F : L C m satisfies H + 12 F ⊥ = 0 . (8)Lagrangian submanifolds in C m satisfying (8) are called shrinking soliton for LMCF. In-deed, the homothetically shrinking family about the origin { F t = √ − tF } t< satisfiesLMCF with F = F . 6 ii) Similarly, the f -minimal Lagrangian submanifolds in C m with f ( z ) := − | z | satisfies H − F ⊥ = 0 . (9)Lagrangian submanifolds in C m satisfying (9) are called expanding soliton for LMCF.Indeed, the homothetically expanding family about the origin { F t = √ tF } t> − satisfiesLMCF with F = F . (iii) Let T ∈ C m be a fixed vector, define f ( z ) := 2 h z, T i . Then ∇ f = 2 T . The f -minimalLagrangian submanifolds in C m satisfies H + T ⊥ = 0 . (10)Lagrangian submanifolds in C m satisfying (10) are called translating soliton for LMCF.Indeed, the family moving by translation in T -direction { F t = F − tT } t ∈ R satisfies LMCFwith F = F . f -minimal LagrangianSubmanifolds First we introduce the differential operators that will be used in the following sections. Let(
X, J, ω, f ) be a K¨ahler manifold with a real holomorphic potential and F : L X be aLagrangian submanifold with induced metric g . To simplify the notations, we will continue touse f to denote the restriction of the ambient function f to L . Consider the space of weighted L differential forms L (Λ ∗ T ∗ L, e − f dV g ) on L with inner product( · , · ) f := Z L h · , · i g e − f dV g . Then the formal adjoint of d with respect to ( · , · ) f is given by d ∗ f := d ∗ g + ι ∇ f . Define∆ f := d d ∗ f + d ∗ f d, then ∆ f is a positive-definite self-adjoint operator with respect to ( · , · ) f . This operator is usuallycalled the Witten Laplacian associated to f .We are now ready to prove the second variation formula for V f . Theorem 3.1.
Assume that ( X, J, ω, f ) is a K¨ahler manifold with a real holomorphy potential,and F : L X is an f -minimal Lagrangian submanifold. Then for any compactly supportednormal variation { F t } t ∈ ( − ǫ,ǫ ) with F = F and ddt (cid:12)(cid:12) t =0 F t = ξ ∈ Γ c ( N L ) , we have ( δ V f ) F ( L ) ( ξ ) = Z L { h e ω − ◦ ∆ f ◦ e ω ( ξ ) , ξ i − [ Ric( ξ, ξ ) + Hess f ( ξ, ξ ) ] } e − f dV g . (11) Proof.
Differentiate (6) again and use the f -minimal condition H + ( ∇ f ) ⊥ = 0, d dt V f ( L t ) (cid:12)(cid:12) t =0 = − Z L [ h ∇ ξ ( H + 12 ∇ f ) , ξ i + h ∇ f, ∇ ξ ξ i ] e − f dV g .
7y the same computation as in the second variation formula for minimal submanifolds, h ∇ ξ H, ξ i = h ∆ N ξ − R ( ξ ) + ˜ A ( ξ ) , ξ i , where h R ( ξ ) , ξ i = P mi =1 h R ( e i , ξ ) e i , ξ i for any orthonormal basis { e i } on L and ˜ A = A t A for A denoting the second fundamental form. We also compute h ∇ ξ ∇ f, ξ i = Hess f ( ξ, ξ ) , and h ∇ f, ∇ ξ ξ i = −h ∇ ξ ∇ f, ξ i = −h ∇ ∇ f ξ, ξ i , where in the last equality we use the fact that h [ ξ, ∇ f ] , ξ i = 0. Combining these three termswe get d dt V f ( L t ) (cid:12)(cid:12) t =0 = Z L [ h ( − ∆ N ξ + R ( ξ ) − ˜ A ( ξ )) + 12 ∇ ∇ f ξ, ξ i −
12 Hess f ( ξ, ξ ) ] e − f dV g . By Gauss formula and f -minimality, given any orthonormal basis { e i } on L , h R ( ξ ) , ξ i = − m X i =1 h R ( e i , ξ ) ξ, e i i = − Ric( ξ, ξ ) + m X i =1 h R ( J e i , ξ ) ξ, J e i i = − Ric( ξ, ξ ) + m X i =1 h R ( e i , J ξ ) J ξ, e i i = − Ric( ξ, ξ ) + m X i =1 h R ( e i , J ξ ) J ξ, e i i + h B ( e i , J ξ ) , B ( e i , J ξ ) i − h B ( e i , e i ) , B ( J ξ, J ξ ) i = − Ric( ξ, ξ ) + Ric(
J ξ, J ξ ) + h ˜ Aξ, ξ i − h ∇ Jξ ( ∇ f ) ⊥ , J ξ i . On the other hand, we have the following relation proved by Oh ([27], Lem.3.3):
Lemma 3.2.
For any ξ ∈ Γ( N L ) we have(i) ∇ X ( e ω ( ξ )) = e ω (( ∇ X ξ ) ⊥ ) for all X ∈ Γ( T L ) , and(ii) ∆ N ξ = e ω − ◦ ∆ ◦ e ω ( ξ ) , where ∆ is the covariant Laplacian acting on Γ( T ∗ L ) . By Lemma 3 . N ξ = − e ω − ◦ (∆ h − Ric) ◦ e ω ( ξ ) , where ∆ h = dd ∗ g + d ∗ g d is the Hodge Laplacian. Now the f -Laplacian acting on 1-forms on L isgiven by ∆ f = d d ∗ f + d ∗ f d = ∆ h + 12 L ∇ f ,
8o expressing the Lie derivative by covariant derivative we get h e ω − ◦ (∆ h − Ric) ◦ e ω ( ξ ) , ξ i = h e ω − ◦ (∆ f − L ∇ f − Ric) ◦ e ω ( ξ ) , ξ i = h e ω − ◦ (∆ f − ∇ ∇ f ) ◦ e ω ( ξ ) , ξ i − h ∇ Jξ ∇ f, J ξ i − Ric(
J ξ, J ξ )= h e ω − ◦ ∆ f ◦ e ω ( ξ ) − ∇ ∇ f ξ, ξ i − h ∇ Jξ ∇ f, J ξ i − Ric(
J ξ, J ξ ) , where in the third line we use the Lemma 3 . e ω − ◦ ∇ ∇ f ◦ e ω = ( ∇ ∇ f ξ ) ⊥ .Combining everything together we finally obtain d dt V f ( L t ) (cid:12)(cid:12) t =0 = Z L [ h e ω − ◦ ∆ f ◦ e ω ( ξ ) , ξ i − Ric( ξ, ξ ) −
12 Hess f ( ξ, ξ ) −
12 Hess f ( J ξ, J ξ ) ] e − f dV g . Notice that, by the assumption on f ,12 Hess f ( ξ, ξ ) + 12 Hess f ( J ξ, J ξ ) = Hess f ( ξ, ξ ) . Now we can define the notions of f -stability for f -minimal Lagrangians. Definition 3.3.
Let ( X, J, ω, f ) be a K¨ahler manifold with a real holomorphy potential. An f -minimal Lagrangian F : L X is called(i) f-stable if ( δ V f ) F ( L ) ( ξ ) ≥ for all ξ ∈ Γ c ( N L ) ,(ii) Lagrangian f-stable if ( δ V f ) F ( L ) ( ξ ) ≥ for ξ ∈ Γ c ( N L ) with e ω ( ξ ) being closed, and(iii) Hamiltonian f-stable if ( δ V f ) F ( L ) ( ξ ) ≥ for ξ ∈ Γ c ( N L ) with e ω ( ξ ) being exact. By the same proof as in [27] Theorem 3.6 and Theorem 4.4, we have
Corollary 3.4.
Let ( X, J, ω, f ) be a K¨ahler manifold with a real holomorphy potential withBakry– ´Emery Ricci curvature Ric f := Ric + Hess f .(i) If Ric f ≤ , then any f -minimal Lagrangian is f -stable.(ii) If Ric f = c · g with c > , then any f -minimal Lagrangian is Hamiltonian f -stable if andonly if λ (∆ f ) ≥ c . Notice that if we take f to be a constant, this corollary reduces to Oh’s original results.From Example 2 and Theorem 3 .
1, we obtain the f -stability for LMCF solitons. Corollary 3.5. (i) Every Lagrangian expanding soliton is strictly f -stable.(ii) Every Lagrangian translating soliton is f -stable. Remark 1.
It is known that shrinking solitons for MCF are f -unstable, so one has to considerstability with respect to the “entropy” defined by Coding–Minicozzi [7], called the F -stability .See [18] for some F -stability criterions for closed Lagrangian shrinking solitons.9 Calibrated Submanifolds with respect to the f -volume f -special Lagrangian Submanifolds Recall that Harvey and Lawson [10] shows that if (
X, J, ω,
Ω) is a Calabi–Yau m -fold, then forany Lagrangian submanifold F : L X we have F ∗ Ω = e iθ dV g for some θ : L → R / π Z , called the Lagrangian angle . The mean curvature vector satisfies H = J ∇ θ , thus if F : L X is minimal, then θ = θ is a constant. Moreover, in thiscase F : L X is calibrated by Re( e − iθ Ω) and hence it is actually volume-minimizing in itshomology class.We now generalize the above theory to Lagrangian submanifolds in gradient steady KRsolitons, and give an alternative description of f -stability for f -minimal Lagrangians. Given agradient steady KR soliton ( X, J, ω, f ), Robert Bryant [4] shows that there exists a nonvanishingholomorphic volume form, denoted by Ω f , such that e − f ω m m ! = ( − m ( m − (cid:18) i (cid:19) m Ω f ∧ Ω f . (12)In other words, ( X, J, ω, Ω f ) is almost Calabi–Yau in the sense of Joyce (see [14], Def. 8.4.3).Define e g := e − fm g , then for any θ ∈ R / π Z , Re ( e − iθ Ω f ) is a calibration with respect to theconformal metric e g . We rephrase this from the view point of the f -volume. Definition 4.1. (i) A p -form α on ( X, g, f ) is called an f -calibration if dα = 0 and α (cid:12)(cid:12) P ≤ e − p m f vol P for any p -dimensional oriented subspace P ⊂ T x X , for all x ∈ X .(ii) A p -submanifold F : L X in X is said to be f -calibrated by an f -calibration α if F ∗ α = e − p m F ∗ f dV g , where dV g is the induced volume on L . It is not hard to see that any f -calibrated submanifold is f -minimal and any compact f -calibrated submanifold minimizes the f -volume in its homology class. One can show thatRe ( e − iθ Ω f ) is an f -calibration and the f -calibrated submanifolds are f -minimal Lagrangiansubmanifolds. Conversely, by choosing an orientation, any f -minimal Lagrangian submanifoldsin a gradient steady KR soliton is f -calibrated by Re ( e − iθ Ω f ) for some θ ∈ R / π Z . Definition 4.2.
We call the submanifolds calibrated by
Re ( e − iθ Ω f ) f-special Lagrangian sub-manifolds ( f -SLag for short) with phase θ . If F is Lagrangian, then by the same method as in [10] one can show that F ∗ Ω f = e iθ e − F ∗ f dV g for some θ ∈ R / π Z . We still call θ the Lagrangian angle . It turns out that F : L X is an f -SLag with phase θ if and only if F : L X is Lagrangian with constantLagrangian angle θ Remark 2.
In Joyce’s terminology (see [9], Def. 8.4.4), f -SLags in our sense are still called special Lagrangians . We put an f here to emphasize the role of the real holomorphy potentialand the relation to f -minimal submanifolds. 10e now give a family of examples of f -SLags in C m . Example 3 (Lagrangian Translating Solitons) . Consider a Lagrangian translating soliton F : L C m with Lagrangian angle θ . Then as in [26], H + T ⊥ = 0 ⇐⇒ J ∇ θ + T ⊥ = 0 ⇐⇒ ∇ θ − ( J T ) T = 0 ⇐⇒ ∇ θ − ∇h F, J T i = 0 . So F satisfies the translator equation θ ( F ) − h F, J T i = θ (13)for some constant θ ∈ R . We shall show that Lagrangian translating solitons are f -calibratedwith phase θ .Let Ω := dz ∧ · · · ∧ dz m and Ω f := e − f − i h z,JT i Ω , where f ( z ) = 2 h z, T i . Then Ω f is a holomorphic volume form on C m and satisfies (12). Hence( C m , i, ω , Ω f ) is almost Calabi–Yau. By Lagrangian condition we then have F ∗ (Ω f ) = e i ( θ ( F ) −h F,JT i ) e − F ∗ f dV g = e iθ ( e − F ∗ f dV g ) . Therefore Lagrangian translating solitons are f -SLag with phase θ .When f = 0, f -SLags reduce to SLags in Calabi–Yau manifolds. Hence f -minimal La-grangian submanifolds in gradient steady KR solitons are generalizations of special Lagrangiansin Calabi–Yau manifolds. From Example 3, they can also be considered as generalizations of La-grangian translating solitons in C m . In fact, under MCF, f -SLags are evolved by “translation”along the flow of the vector field − ∇ f on X (see section 5 . f -SLags. The equation for f -SLag defor-mation vector fields is given by the next lemma. Lemma 4.3.
Let ( X, J, ω, f ) be a steady KR soliton and F : L → X be an f -SLag. Then the f -SLag deformation vector fields ξ ∈ Γ( N L ) is characterized by the f -harmonic -form equation d e ω ( ξ ) = d ∗ f e ω ( ξ ) = 0 . (14) Proof.
Without loss of generality, we may assume F has phase 0. Let { F t } be a family ofimmersions satisfies F = F and ddt (cid:12)(cid:12) t =0 F t = ξ ∈ Γ( N L ). Then ξ preserves f -SLag condition ifand only if ddt (cid:12)(cid:12) t =0 F ∗ t ω = ddt (cid:12)(cid:12) t =0 F ∗ t Im(Ω f ) = 0 . It is well known that ddt (cid:12)(cid:12) t =0 F ∗ t ω = 0 if and only if d e ω ( ξ ) = 0. Then we compute ddt (cid:12)(cid:12) t =0 F ∗ t Ω f = F ∗ L ξ Ω f = F ∗ ( dι ξ Ω f )= − i d ι Jξ F ∗ Ω f = − i d ( ι Jξ e − F ∗ f dV g )= − i ( − d ( F ∗ f ) e − F ∗ f ∧ ι Jξ dV g + e − F ∗ f d ι Jξ dV g )= i ( d ∗ g e ω ( ξ ) + 12 ι ∇ F ∗ f e ω ( ξ )) e − F ∗ f dV g . ddt (cid:12)(cid:12) t =0 F ∗ t Im(Ω f ) = 0 if and only if d ∗ g e ω ( ξ ) + 12 ι ∇ F ∗ f e ω ( ξ ) = d ∗ f e ω ( ξ ) = 0 . Notice that (14) also appears in the second variation formula for f -volume (Theorem 3 .
1) asthe Jacobi field equation for f -minimal Lagrangians.From Lemma 4 .
3, for compact f -SLags, the deformation theory is the same as special La-grangians, as shown by the following theorem: Theorem 4.4 ([9], Thm.10.8) . Let ( X, J, ω, Ω f ) be an almost Calabi–Yau m -fold and F : L m X be a compact f -SLag submanifold. Then the moduli space of f -SLags is a smooth manifold ofdimension b ( L ) . Unfortunately, just like minimal submanifolds in Euclidean space, we have the followingnoncompact result for f -minimal Lagrangians. Proposition 4.5.
Let ( X, J, ω, f ) be a gradient steady or expanding K¨ahler–Ricci soliton whichis not K¨ahler-Einstein, and F : L X be an f -minimal Lagrangian submanifold. Then L mustbe noncompact.Proof. Let
X, Y be tangent vectors of L . Then by f -minimality,Hess f ( X, Y ) = h∇ X ∇ f, Y i = h∇ X ∇ f, Y i + h∇ X ( ∇ f ) ⊥ , Y i = Hess f ( X, Y ) − h∇ X H, Y i [ Since H + ( ∇ f ) ⊥ = 0 . ]= Hess f ( X, Y ) + 2 A ( H, X, Y ) . We first deal with the steady case. Since Ric + Hess f = 0,Ric( X, Y ) = − Hess f ( X, Y ) − A ( H, X, Y ) . Let { e i } mi =1 be an orthonormal basis of T p L for some p ∈ L . Then m X i =1 Ric( e i , e i ) = − ∆ f − | H | = − ∆ f + 12 |∇ f | − |∇ f | [ Since ∇ f = ∇ f − H. ]Since F : L X is Lagrangian, { e i , J e i } mi =1 is an orthonormal basis of T p X , Hence we have R = m X i =1 Ric( e i , e i ) + m X i =1 Ric(
J e i , J e i ) = 2 m X i =1 Ric( e i , e i ) . (15)Combining these equations we obtain R + |∇ f | = − f + |∇ f | . R + |∇ f | is constant on X (see also [4] for a differentproof). Therefore f satisfies ∆ f − |∇ f | = c for some c ∈ R on L . The result then follows from maximum principle.Next we prove the expanding case. We may assume Ric + Hess f = − g . For any vectors X, Y tangent to L we haveRic( X, Y ) + Hess f ( X, Y ) + 2 A ( H, X, Y ) = − g ( X, Y ) . Taking the tangential trace on both sides and use (15) we get∆ f − |∇ f | + 12 |∇ f | + R − m. (16)On any gradient expanding soliton we know that (see [22]), after adding a suitable constant to f , R + |∇ f | + 2 f = 0. Hence f satisfies∆ f − |∇ f | − f = − m. Assume x ∈ L is a local minimum of f restricted on L , then0 ≤ ∆ f ( x ) = − m + f ( x ) = ⇒ f (cid:12)(cid:12) L ≥ m. But from [39] Corollary 2.4, the scalar curvature is bounded from below R ≥ − m , so2 f = − R − |∇ f | ≤ − R ≤ m = ⇒ f ≤ m on X. Therefore if f (cid:12)(cid:12) L attains minimum on L , then f (cid:12)(cid:12) L ≡ m is constant on L . Hence R (cid:12)(cid:12) L = − m ,which means the minimum of R is attained on L . By [39] Corollary 2.4, g is Einstein, acontradiction. We consider the special case that F : L C m is a Lagrangian translating soliton. Let( x , · · · , x m , y , · · · , y m ) be standard coordinates in R m ≃ C m . The Liouville form is definedby λ := P mi =1 y i dx i − x i dy i . A Lagrangian submanifold F : L C m is said to be exact if F ∗ λ = dβ for some β ∈ C ∞ ( L ). The exact deformations (deformations that preserves exactness) areinduced by exact 1-forms on L , that is, ξ ∈ Γ c ( N L ) is an exact deformation if and only if e ω ( ξ )is exact (see [20], Lemma 5.4).We will restrict our attention to the study of exact deformations of an exact Lagrangiantranslating soliton F : L C m . In this case, (14) reduces to the f -Laplace equation∆ f u = 0 (17)on L , where ∆ f = − d ∗ f d = ∆ − h∇ f, · i is the Witten Laplacian acting on functions. Thesolutions u to (17) are called f -harmonic functions on L . We show that there is a weighted L gap between f -harmonic functions. 13 roposition 4.6. Suppose F : L C m is a Lagrangian translating soliton with H + T ⊥ = 0 , | T | = 1 . Let f ( z ) = 2 h z, T i and u be a function on L satisfying R L u e − f dV g < ∞ and ∆ f u = 0 .Then u ≡ .Proof. Suppose ∆ f u = 0. Let w := u , then∆ f w = 2 u ∆ u + 2 |∇ u | − u h∇ u, ∇ f i = 2 |∇ u | . Fix p ∈ L , consider a sequence of cut-off functions { φ R } R> satisfying φ R = 1 in B R ( p ) , φ R = 0 outside B R ( p ) , and |∇ φ R | ≤ CR for some C >
0. Then0 = Z L div L ( e − f φ R ∇ u ) dV g = Z L h∇ ( e − f φ R ) , ∇ u i dV g + Z L e − f φ R ∆ u dV g = Z L h u ∇ φ R , φ R ∇ u i e − f dV g + Z L |∇ u | φ R e − f dV g . By Young’s inequality with ǫ = 1 / Z L |∇ u | φ R e − f dV g = − Z L h u ∇ φ R , φ R ∇ u i e − f dV g ≤ Z L u |∇ φ R | e − f dV g + Z L φ R |∇ u | e − f dV g . Thus letting R → ∞ , by finiteness of R L u e − f dV g we obtain R L |∇ u | e − f dV g = 0, hence u isconstant.To show u = 0, it is enough to show that L has infinite weighted volume. First notice thatwe have the identities ∆ f = − | H | and | H | + 14 |∇ f | = 1 , so ∆ f e f = 14 e f (cid:18) |∇ f | − (cid:19) ≤ − e f . From this we deduce that λ (∆ f ) ≥ (see, for example, Proposition 22.2 of [19]). Then by asimple argument in [34] Corollary 4.2, we conclude that L has infinite weighted volume.From [26] proposition 2.2, we have ∆ f θ = 0 on Lagrangian translating solitons. Thus θ is f -harmonic. This corresponds to the fact that the exact deformation vector field induced by θ is just the mean curvature H = J ∇ θ , which is just a translation in C m . Corollary 4.7. If F : L C m is a Lagrangian translating soliton as in Proposition . , u ∈ C ∞ ( L ) with ∆ f u = 0 and R L ( u − θ ) e − f dV g < ∞ , then u ≡ θ . u has finite weighted L distance to the Lagrangian angle θ . This provides a kind ofinfinitesimal uniqueness of exact Lagrangian translating solitons.From f -harmonicity of θ , we also have a nonexistence theorem in 2-dimensions. Corollary 4.8. If F : L C m is a Lagrangian translating surface as in Proposition . withLagrangian angle θ satisfying R L θ e − f dV g < ∞ , then L is a plane.Proof. By proposition 4 . θ ≡ L , so T ⊥ = − H = 0, that is, T is tangent to L . Therefore L ≃ Σ × R ⊂ C × C for some minimal curve Σ. Hence Σ is a line and L is a plane. Suppose now (
X, J, ω ) is a K¨ahler manifold and f is a smooth function which is not necessarilya holomorphy potential. Then by the same computations as the proof of Theorem 3 .
1, one canshow that the second variation formula of f -volume becomes( δ V f ) F ( L ) ( ξ ) = Z L { h e ω − ◦ ∆ f ◦ e ω ( ξ ) , ξ i − [ ρ ( ξ, J ξ ) + i ∂∂f ( ξ, J ξ ) ] } e − f dV g , (18)where ρ = Ric( J · , · ) is the Ricci form of ω . In this case, the f -stability depends on the bilinearform ρ ( · , J · ) + i ∂∂f ( · , J · ). In particular, if ( X, J, ω, f ) is almost-Einstein , that is, ρ + i ∂∂f = C ω (19)for some C ∈ R , then ρ ( · , J · ) + i ∂∂f ( · , J · ) = C g ( · , · ). Thus Corollary 5.1.
Suppose ( X, J, ω, f ) is almost-Einstein with ρ + i ∂∂f = C ω for some C ∈ R .Then(i) every f -minimal Lagrangian submanifold is f -stable if C ≤ , and(ii) if C > , any compact f -minimal Lagrangian submanifold is Hamiltonian f -stable if andonly if λ (∆ f ) ≥ C . Notice that the above Hamiltonian f -stability criterion is also obtained in [15]. A longstanding problem in Geometry is the existence problem for SLags in Calabi–Yau man-ifolds. Since SLags are volume minimizing, one approach to tackle the existence problem isto deform an initial Lagrangian submanifold along the negative gradient flow of the volumefunctional, namely, the mean curvature flow (MCF) (cid:18) ddt F t (cid:19) ⊥ = H ( F t ) . Smoczyk [29] proves that the Lagrangian condition is preserved by MCF if the ambient spaceis K¨ahler-Einstein, and in this case the flow is called Lagrangian mean curvature flow (LMCF).However, finite-time singularities often occur and therefore in general one cannot have long-time15xistence and convergence. There are conjectural pictures in dealing with this problem, see forexample, Thomas-Yau [31] and Joyce [13].A relevant question about long-time existence and convergence of LMCF one can ask is therelation between stability of minimal Lagrangians under volume functional and dynamic stability of LMCF, that is, whether a small Lagrangian perturbation of a stable minimal Lagrangiansubmanifold converges back to the original minimal submanifold along LMCF? Results in thisdirection can be found in, for instance, Li [17], Tsai-Wang [33], [32], see also Lotay-Schulze [21]for an applications of [32] to LMCF with singularities.The above picture can be generalized to f -minimal Lagrangians. More precisely, we considerthe negative gradient flow of the f -volume functional: ddt F t = H ( F t ) + 12 ( ∇ f ) ⊥ g, Ft . (20)Behrndt [3] shows that if ( X, J, ω, f ) is almost-Einstein, then the Lagrangian condition is pre-served by the flow (20), called the generalized Lagrangian mean curvature flow (GLMCF) . Thestationary points of (20) are the f -minimal Lagrangians. Therefore we can ask the same ques-tion about dynamic stability of GLMCF. Kajigaya–Kunikawa [15] recently generalized Li’s result[17] and obtained a dynamic stability theorem for compact f -minimal Lagrangians in compactalmost-Einstein K¨ahler manifolds. Besides the compact cases, the dynamic stability for LMCFsolitons under GLMCF are especially interesting since in this case the GLMCF corresponds toLMCF with scalings. Problem 1.
Are the expanding and translating solitons for LMCF dynamically stable underGLMCF?
The author believe that this problem is related to the conjectural theory of formation anddesingularization of singularities of LMCF proposed by Joyce [13].
There is another generalization of LMCF by considering the mean curvature flow along a movingambient metric. Let { g ( t ) } be a solution to KRF, that is, its K¨ahler form ω ( t ) satisfies ddt ω ( t ) = − ρ ( ω ( t )) , for t ∈ ( a, b ) , (21)where ρ denotes the Ricci form. We consider the mean curvature flow { F t } along { g ( t ) } , thatis, (cid:18) ddt F t (cid:19) ⊥ g ( t ) ,Ft = H ( F t ) for t ∈ ( a, b ) , (22)where the mean curvature H ( F t ) of F t is computed with respect to g ( t ). The couple ( g ( t ) , F t )defined by (21) and (22) is called the K¨ahler–Ricci mean curvature flow (KR-MCF for short).Smoczyk [29] shows that Lagrangian condition is preserved by KR–MCF.Now, if we are given a gradient KR soliton ( X, J, ω, f ), then there is a canonical KRFsolution g ( t ) := σ ( t ) ϕ ∗ t g, (23)which is defined for all t such that σ ( t ) > ϕ t is the biholomorphism on X generated by σ ( t ) ∇ f . In this case there is an one-to-one correspondence between GLMCFand KR-MCF, as shown in the following lemma.16 emma 5.2. Let ( X, J, ω, f ) be a gradient KR soliton and let g ( t ) be the solution to KRF definedas above. If ( g ( t ) , C t ) is the solution to KR-MCF for t ∈ ( a, b ) , we set F s ( t ) := ϕ t ◦ C t for s ( t ) = R ta dτσ ( τ ) . Then F s : L X satisfies the generalized LMCF in the fixed background ( X, J, ω ) .Conversely, given a generalized LMCF { F s } in ( X, J, ω ) , then ( g ( t ) , C t := ( ϕ t ) − ◦ F s ( t ) ) satisfiesthe KR-MCF.Proof. Let s ( t ) to be determined. We compute dds F s = dtds ∂∂t ( ϕ t ◦ C t ) = dtds (cid:20) σ ( t ) ( ∇ f ◦ ϕ t ◦ C t ) + dϕ t ( H ( C t )) (cid:21) . By solving dtds = σ ( t ), we obtain s ( t ) = R ta dτσ ( τ ) . Then by taking the normal component withrespect to g and F s , (cid:18) dds F s (cid:19) ⊥ g, Fs = 12 ( ∇ f ) ⊥ g, Fs + σ ( t ) dϕ t ( H ( C t ))= 12 ( ∇ f ) ⊥ g, Fs + H ( F s ) . The converse follows from similar calculations.Notice that the case for shrinking solitons in shrinking Ricci solitons has been proved by Ya-mamoto [36], [37].If we put F s = F : L X to be an f -minimal Lagrangian, then the KR-MCF evolves F by C t = ( ϕ t ) − ◦ F , defined for all t such that σ ( t ) = 1 − ct >
0. Therefore we conclude that
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