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Hamiltonian L-stability of Lagrangian Translating Solitons

In this paper, we compute the first and second variation formulas for the F-functional of translating solitons and study the Hamiltonian L-stability of Lagrangian translating solitons. We prove that any Lagrangian translating soliton is Hamiltonian L-stable.


Introduction
An n-dimensional submanifold Σ n of R n+p is called a self-shrinker if it is the time t = −1 slice of a self-shrinking mean curvature flow that disappears at (0, 0), i.e. of a mean curvature flow satisfying Σ t = √ −tΣ −1 . We can also consider a self-shrinker as a submanifold that satisfies An n-dimensional submanifold Σ n of R n+p is called a translating soliton if there is a constant vector T so that Σ t = Σ + tT is a solution to the mean curvature flow. We can also consider a translating soliton as a submanifold that satisfies According to the blow up rate of the second fundamental form, Huisken [6] classified the singularities of mean curvature flows into two types: Type I and Type II. Any Type I singularity of the mean curvature flow must be a self-shrinker ( [6]). Type II singularity is one class of eternal solutions, which is defined for −∞ < t < ∞. One of the most important example of Type II singularity is the translating soliton ( [5,7]).
In this paper, we mainly study the stability (in some sense) of translating solitons. It was motivated by the work of Colding-Minicozzi [4], where they introduced the concept of F-stability of a self-shrinker in the hypersurface case. The definitions of many concepts in their paper can be naturally generalized to the higher codimension case (cf. [1,2,8]). Given x 0 ∈ R n+p and t 0 > 0, F x 0 ,t 0 is defined by In [4], Colding-Minicozzi proved that self-shrinkers are the critical points for the F 0,1 functional by computing the first variation formula of F 0,1 . They also computed the second variation formula, and defined F-stability of a self-shrinker by modding out translations. They showed that the round sphere and hyperplanes are the only F-stable self-shrinkers in R n+1 . In 2002, Andrews-Li-Wei [1], Arezzo-Sun [2] and Lee-Lue [8] independently generalized some of Colding-Minicozzi's work [4] from the hypersurface case to the higher codimensional case. They computed the first and second variation formulas for the F-functional, and studied F-stability of self-shrinkers in higher codimension.
Recently, motivated by an observation by Oh [10], Li-Zhang [9] and the author [12] studied Lagrangian F-stability and Hamiltonian F-stability of Lagrangian self-shrinkers, and proved characterization theorems for Hamiltonian F-stability of Lagrangian self-shrinkers, which characterize the Hamiltonian F-stablity by the eigenvalues and eigenspaces of the drifted Laplacian.
With the above known results for self-shrinkers, it is natural to think that translating solitons might also have some similar properties. In fact, translating solitons are also critical points for an F -functional, which was studied by some people (See [3], [11] and [13] for example ). The F -functional is defined by It is not as good as in the self-shrinker case because here F (Σ) is usually infinity if Σ is a translating soliton, since any translating soliton is noncompact and e T,x → ∞ very quickly as x → ∞. This makes it hard to get many corresponding results as in the self-shrinker case.
However, if we require variation vector fields to have compact support, we can still compute the first and second variation formulas and consider stability of translating solitons. In [11], Shahriyari defined L-stability of translating surfaces in R 3 , and proved that any translating graph in R 3 is L-stable.
Especially, if a translating soliton is also a Lagrangian submanifold of the Euclidean space, we call it a Lagrangian translating soliton. We consider Hamiltonian L-stability (see section 2 for the definition) of Lagrangian translating solitons. Our main theorem is Theorem 1.1. Any Lagrangian translating soliton is Hamiltonian L-stable.
Since we have this theorem, an interesting question is that whether this theorem has some application that could help us study the Lagrangian mean curvature flow. Besides, we are also interested in the L-stability of translating solitons in the hypersurface case.
Acknowledgement: The author would like to thank Professor William P. Minicozzi II for introducing this problem to her and for helpful suggestions. She would also like to thank Professor Gang Tian for his support and comments.

Variation formulas and Hamiltonian L-stability
2.1. First and second variation formulas. Recall that the F-functional is defined by The first variation formula of F is Proof. From the first variation formula (for area), we know that This proves the lemma. Q.E.D. It follows that The second variation formula at a critical point is Proof. Letting primes denote derivatives with respect to s at s = 0, differentiating (2.1) gives Similar to the derivation of the second variation formula for the area, we have On the other hand, since V, T T , V = 0, it follows that The next corollary is an extension of Lemma 2.4, which follows immediately by choosing cut-off functions and using the dominated convergence theorem, the same as in the proof of Corollary 3.10 in [4].
Now we calculate some equalities that we think will be useful in the future.

Besides, we have
Proof. Fix p ∈ Σ and choose an orthonormal frame {e i } such that ∇ e i e j (p) = 0, g ij = δ ij in a neighborhood of p. We have Especially, choosing y = T , we have where we used (2.10), ∇ e k e j (p) = 0, and the Codazzi equation in the last equality. Taking the trace of (2.13) and using H = T ⊥ , we conclude that This proves (2.7).

Proof of the main theorem
Note that the normal bundle brings much difficulty to the study of L-stability of translating solitons in the general higher codimension case. However, in [10], Oh studied Hamiltonian stability of minimal Lagrangian submanifolds in Kähler-Einstein manifolds, and characterized Hamiltonian stability by a condition on the first eigenvalue of ∆ acting on functions. The key point of Oh's proof is that, for a minimal Lagrangian submanifold of a Kähler-Einstein manifold, the set of Hamiltonian variations is an invariant subspace of the Jacobi operator. This idea was then used to study Hamiltonian F-stability of Lagrangian self-shrinkers ( [9], [12]). It is natural to think that this propery also holds for Lagrangian translating solitons. This property inspired us to show the following equality, which well characterizes how the operator L acts on Hamiltonian variations. Proof. Fix a point p. We choose a local orthonormal basis {e i } n i=1 of T Σ such that ∇ e i e j (p) = 0. Then since Σ is Lagrangian, {e n+i = Je i } n i=1 is a local orthonomal basis of N Σ. In the following we compute at the point p. It is easy to compute that where in the last equality we used the Lagrangian property h n+k il = h n+l ik . On the other hand,  where the last equality used Lemma 2.4 and the fact that V = J∇f is compactly supported implies that Lf is compactly supported. This proves the theorem. Q.E.D.