Complete Self-Shrinking Solutions for Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space

Abstract

Let be a smooth strictly convex solution of defined on a domain ; then the graph of is a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean space with the indefinite metric . In this paper, we prove a Bernstein theorem for complete self-shrinkers. As a corollary, we obtain if the Lagrangian graph is complete in and passes through the origin then it is flat.

1. Introduction

Let be an -dimensional submanifold immersed into the Euclidean space . Mean curvature flow is a one-parameter family of immersions with corresponding images such that is satisfied, where is the mean curvature vector of at in . Self-similar solutions to the mean curvature flow play an important role in understanding the behavior of the flow and the types of singularities. They satisfy a system of quasilinear elliptic PDE of the second order as follows: where stands for the orthogonal projection into the normal bundle .

Self-shrinkers in the ambient Euclidean space have been studied by many authors; for example, see [1–6] and so forth. For recent progress and related results, see the introduction in [7]. When the ambient space is a pseudo-Euclidean space, there are many classification works about self-shrinkers; for example, see [8–13] and so forth. But very little is known when self-shrinkers are complete not compact with respect to induced metric from pseudo-Euclidean space. In this paper, we will characterize self-shrinkers for Lagrangian mean curvature flow in the pseudo-Euclidean space from this aspect.

Let be null coordinates in 2-dimensional pseudo-Euclidean space . Then, the indefinite metric (cf. [14]) is defined by . Suppose is a smooth strictly convex function defined on domain . The graph of can be written as . Then, the induced Riemannian metric on is given by In particular, if function satisfies then the graph of is a space-like self-shrinking solution for mean curvature flow in .

Huang and Wang [12] and Chau et al. [8] have used different methods to investigate the entire solutions to the above equation and showed that an entire smooth strictly convex solution to (4) in is the quadratic polynomial under the decay condition on Hessian of . Later Ding and Xin in [10] improve the previous ones in [8, 12] by removing the additional assumption and prove the following.

Theorem 1. Any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in with the indefinite metric is flat.

These rigidity results assume that the self-shrinker graphs are entire. Namely, they are Euclidean complete. Here, we will characterize the rigidity of self-shrinker graphs from another completeness and pose the following problem.

If a graphic self-shrinker is complete with respect to induced metric from ambient space , then is it flat?

In this paper, we will use affine technique (see [15–18]) to prove the following Bernstein theorem. As a corollary, it gives a partial affirmative answer to the above problem.

Theorem 2. Let be a strictly convex function defined on a convex domain satisfying the PDE (4). If there is a positive constant depending only on such that the hypersurface in is complete with respect to the metric then is the quadratic polynomial.

Remark 3. If is a strictly convex solution to (4), then the graph is a minimal manifold in endowed with the conformal metric .

As a direct application of Theorem 2, we have the following.

Corollary 4. Let be a strictly convex -function defined on a convex domain . If the graph in is a complete space-like self-shrinker for mean curvature flow and the sum has a lower bound, then is flat.

When the shrinker passes through the origin especially, we have the following corollary.

Corollary 5. If the graph in is a complete space-like self-shrinker for mean curvature flow and passes through the origin, then is flat.

2. Preliminaries

Let be a strictly convex -function defined on a domain . Consider the graph hypersurface For , we choose the canonical relative normalization . Then, in terms of the language of the relative affine differential geometry, the Calabi metric is the relative metric with respect to the normalization . For the position vector , we have where “,” denotes the covariant derivative with respect to the Calabi metric . We recall some fundamental formulas for the graph ; for details, see [19]. The Levi-Civita connection with respect to the metric has the Christoffel symbols The Fubini-Pick tensor satisfies Consequently, for the relative Pick invariant, we have The Gauss integrability conditions and the Codazzi equations read From (12), we get the Ricci tensor Introduce the Legendre transformation of Define the functions here and later the norm is defined with respect to the Calabi metric. From the PDE (4), we obtain That is, Using (17) and (18), we can get Put . From (19), we have By (17), we get and then Using (17) yields Define a conformal Riemannian metric , where is a constant.

Conformal Ricci Curvature. Denote by the Ricci curvature with respect to the metric ; then where “,” again denotes the covariant derivation with respect to the Calabi metric.

Using the above formulas, we can get the following crucial estimates.

Proposition 6. Let be a strictly convex function satisfying PDE (4). Then, the following estimate holds: where

Because its calculation is standard as in [16], we will give its proof in the appendix.

For affine hyperspheres, Calabi in [20] calculated the Laplacian of the Pick invariant . Later, for a general convex function, Li and Xu proved the following lemma in [17].

Lemma 7. The Laplacian of the relative Pick invariant satisfies where “,” denotes the covariant derivative with respect to the Calabi metric.

Using Lemma 7, we get the following corollary. For the proof, see the appendix.

Corollary 8. Let be a strictly convex function satisfying PDE (4); then

3. Proof of Theorem 2

It is our aim to prove ; thus, from definition of , everywhere on . As in [8], by Euler homogeneous theorem, we get Theorem 2.

Denote by the geodesic distance function from with respect to the metric . For any positive number , let . Denote

Lemma 9. Let be a strictly convex -function satisfying the PDE (4). Then, there exist positive constants and , depending only on , such that

Proof. Step 1. We will prove that there exists a constant depending only on such that
To this end, consider the function defined on , where is a positive constant to be determined later. Obviously, attains its supremum at some interior point . We may assume that is a -function in a neighborhood of . Choose an orthonormal frame field on around with respect to the Calabi metric . Then, at , where “,” denotes the covariant derivative with respect to the Calabi metric as before, and we used the fact . Inserting Proposition 6 into (35), we get Combining (34) with (36) and using the Schwarz inequality, we have Choose small enough such that Then, by substituting the three estimates above, we get here and later denotes positive constant depending only on .
Denote . If , from (39), it is easy to complete the proof of the lemma. In the following, we assume that . Now, we calculate the term . Firstly, we will give a lower bound of the Ricci curvature . Assume that For any , by a coordinate transformation, and   hold for . Then, at , Then, using the Schwarz inequality and (22)–(24), we know that at the point If , then Otherwise, Then, the Ricci curvature on is bounded from below by By the Laplacian comparison theorem, we get where denotes the Laplacian with respect to the metric .
Substituting (46) into (39) yields Note that Multiplying by , at both sides of (47), yields Using the Schwarz inequality, we complete Step 1.
Step 2. We will prove that there is a constant depending only on such that Consider defined on , where is the constant in (38). Obviously, attains its supremum at some interior point . Choose an orthonormal frame field on around with respect to the Calabi metric . Then, at , where “,” denotes the covariant derivative with respect to the Calabi metric as before. Inserting Corollary 8 into (53), we get Applying the Schwarz inequality, we have Inserting these estimates into (54) yields here and later denotes different positive constants depending only on .
We discuss two subcases.
Case 1. If then . In this case, Step 2 is complete.
Case 2. Now, assume that Then, . Thus, The rest of the estimate is almost the same as in Step 1. The only difference is to deal with the term . If , then . We can drop this term.
Otherwise, has a uniform upper bound.
Using the same method as in Step 1, we can estimate the term and finally get Then, combining the conclusion of Step 1, we get This completes the proof of Lemma 9.

Proof of Theorem 2. For any point , choose sufficient large constant such that . Then, for all , . Using Lemma 9, we know Now, let , and we have Consequently, This completes the proof of Theorem 2.

4. Appendix

Proof of Proposition 6. Let , and we choose a local orthonormal frame field of the metric around . Then, where we used (20). In the case , it is easy to get, at , Now, we assume that . Choose a local orthonormal frame field of the metric around such that , , for all . Then, where is a constant to be determined later. Applying (20), we obtain An application of the Ricci identity shows that Substituting (68) and (69) into (67), we obtain Note that Then, (70) and (71) together give us Using the Schwarz inequality gives Using and choosing , we get
In the following, we will calculate the terms and . Note that (17) is invariant under an affine transformation of coordinates that preserved the origin. So, we can choose the coordinates such that and , , for all . From (19), we easily obtain Thus, we get By the same method, as deriving (69), we have Note that . Therefore, by (14), (77), (78), and (79), we obtain On the other hand, we have Then, inserting (80) and (81) into (75), we get Using (77), we have One observes that the Schwarz inequality gives Note that by (17) we have Then, inserting these estimates into (82) yields Proposition 6.

Proof of Corollary 8. Now, we will calculate the term . In particular, if satisfies PDE (4), choose the coordinate such that ; then we have Using (17), we have By the Young inequality and the Schwarz inequality, we have Thus, by inserting (88) into Lemma 7, we obtain Corollary 8.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author is supported by Grants (nos. 11101129 and 11201318) of NSFC. Part of this work was done when the first author was visiting the University of Washington and he thanks Professor Yu Yuan and the support of China Scholarship Council. The second author is supported by Grants (nos. U1304101 and 11171091) of NSFC and NSF of Henan Province (no. 132300410141).