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

Abstract

Let f ( x ) be a smooth strictly convex solution of det ( [superscript] ∂ 2 [/superscript] f / ∂ [subscript] x i [/subscript] ∂ [subscript] x j [/subscript] ) = exp ( 1 / 2 ) [superscript] ∑ i = 1 n [/superscript] [subscript] x i [/subscript] ( ∂ f / ∂ [subscript] x i [/subscript] ) - f defined on a domain Ω ⊂ [superscript] R n [/superscript] ; then the graph [subscript] M ∇ f [/subscript] of ∇ f is a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean space [superscript] R n 2 n [/superscript] with the indefinite metric ∑ d [subscript] x i [/subscript] d [subscript] y i [/subscript] . In this paper, we prove a Bernstein theorem for complete self-shrinkers. As a corollary, we obtain if the Lagrangian graph [subscript] M ∇ f [/subscript] is complete in [superscript] R n 2 n [/superscript] and passes through the origin then it is flat.

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Academic Editor:Ljubomir B. Ciric

Department of Mathematics, Henan Normal University, Xinxiang, Henan 453007, China

Received 14 January 2014; Accepted 15 June 2014; 6 July 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let M be an n -dimensional submanifold immersed into the Euclidean space ^{R n + m} . Mean curvature flow is a one-parameter family _{X t} = X ( · , t ) of immersions _{X t} : M [arrow right] ^{R n + m} with corresponding images _{M t} = _{X t} ( M ) such that [figure omitted; refer to PDF] is satisfied, where H ( x , t ) is the mean curvature vector of _{M t} at X ( x , t ) in ^{R n + m} . 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: [figure omitted; refer to PDF] where ( ... ^{) [perpendicular]} stands for the orthogonal projection into the normal bundle N M .

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 ( _{x 1} , ... , _{x n} ; _{y 1} , ... , _{y n} ) be null coordinates in 2 n -dimensional pseudo-Euclidean space ^{R n 2 n} . Then, the indefinite metric (cf. [14]) is defined by d ^{s 2} = ^{∑ i = 1 n} ... d _{x i} d _{y i} . Suppose f ( x ) is a smooth strictly convex function defined on domain Ω ⊂ ^{R n} . The graph _{M ∇ f} of ∇ f can be written as ( _{x 1} , ... , _{x n} ; ∂ f / ∂ _{x 1} , ... , ∂ f / ∂ _{x n} ) . Then, the induced Riemannian metric on _{M ∇ f} is given by [figure omitted; refer to PDF] In particular, if function f satisfies [figure omitted; refer to PDF] then the graph _{M ∇ f} of ∇ f is a space-like self-shrinking solution for mean curvature flow in ^{R n 2 n} .

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 ^{R n} is the quadratic polynomial under the decay condition on Hessian of f . 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 ^{R n 2 n} with the indefinite metric _{∑ i} ... d _{x i} d _{y i} 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 ^{R n 2 n} , 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 f ( x ) be a ^{C ∞} strictly convex function defined on a convex domain Ω ⊆ ^{R n} satisfying the PDE (4). If there is a positive constant α depending only on n such that the hypersurface M = { ( x , f ( x ) ) } in ^{R n + 1} is complete with respect to the metric [figure omitted; refer to PDF] then f is the quadratic polynomial.

Remark 3.

If f ( x ) is a strictly convex solution to (4), then the graph { ( x , ∇ f / 2 n α ) } is a minimal manifold in ^{R n 2 n} endowed with the conformal metric d ^{s 2} = exp ... { - α x · y } d x · d y .

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

Corollary 4.

Let f be a strictly convex ^{C ∞} -function defined on a convex domain Ω ⊂ ^{R n} . If the graph _{M ∇ f} = { ( x , ∇ f ( x ) ) } in ^{R n 2 n} is a complete space-like self-shrinker for mean curvature flow and the sum ∑ ... _{x i} ( ∂ f / ∂ _{x i} ) has a lower bound, then _{M ∇ f} is flat.

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

Corollary 5.

If the graph _{M ∇ f} = { ( x , ∇ f ( x ) ) } in ^{R n 2 n} is a complete space-like self-shrinker for mean curvature flow and passes through the origin, then _{M ∇ f} is flat.

2. Preliminaries

Let f ( _{x 1} , ... , _{x n} ) be a strictly convex ^{C ∞} -function defined on a domain Ω ⊂ ^{R n} . Consider the graph hypersurface [figure omitted; refer to PDF] For M , we choose the canonical relative normalization Y = ( 0,0 , ... , 1 ) . Then, in terms of the language of the relative affine differential geometry, the Calabi metric [figure omitted; refer to PDF] is the relative metric with respect to the normalization Y . For the position vector y = ( _{x 1} , ... , _{x n} , f ( _{x 1} , ... , _{x n} ) ) , we have [figure omitted; refer to PDF] where "," denotes the covariant derivative with respect to the Calabi metric G . We recall some fundamental formulas for the graph M ; for details, see [19]. The Levi-Civita connection with respect to the metric G has the Christoffel symbols [figure omitted; refer to PDF] The Fubini-Pick tensor _{A i j k} satisfies [figure omitted; refer to PDF] Consequently, for the relative Pick invariant , we have [figure omitted; refer to PDF] The Gauss integrability conditions and the Codazzi equations read [figure omitted; refer to PDF] From (12), we get the Ricci tensor [figure omitted; refer to PDF] Introduce the Legendre transformation of f [figure omitted; refer to PDF] Define the functions [figure omitted; refer to PDF] here and later the norm || · || is defined with respect to the Calabi metric. From the PDE (4), we obtain [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF] Using (17) and (18), we can get [figure omitted; refer to PDF] Put τ : = ( 1 / 2 ) ∑ ... ^{f i j} ( _{ρ i} / ρ ) _{f j} . From (19), we have [figure omitted; refer to PDF] By (17), we get [figure omitted; refer to PDF] and then [figure omitted; refer to PDF] Using (17) yields [figure omitted; refer to PDF] Define a conformal Riemannian metric G ~ : = exp ... { α ( f + u ) } G , where α is a constant.

Conformal Ricci Curvature . Denote by _{R ~ i j} the Ricci curvature with respect to the metric G ~ ; then [figure omitted; refer to PDF] 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 f ( _{x 1} , ... , _{x n} ) be a ^{C ∞} strictly convex function satisfying PDE (4). Then, the following estimate holds: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

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 J . Later, for a general convex function, Li and Xu proved the following lemma in [17].

Lemma 7.

The Laplacian of the relative Pick invariant J satisfies [figure omitted; refer to PDF] 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 f ( _{x 1} , ... , _{x n} ) be a ^{C ∞} strictly convex function satisfying PDE (4); then [figure omitted; refer to PDF]

3. Proof of Theorem 2

It is our aim to prove Φ ...1; 0 ; thus, from definition of ρ , [figure omitted; refer to PDF] everywhere on M . As in [8], by Euler homogeneous theorem, we get Theorem 2.

Denote by s ( _{p 0} , p ) the geodesic distance function from _{p 0} ∈ M with respect to the metric G ~ . For any positive number a , let _{B a} ( _{p 0} , G ~ ) : = { p ∈ M |" s ( _{p 0} , p ) ...4; a } . Denote [figure omitted; refer to PDF]

Lemma 9.

Let f be a strictly convex ^{C ∞} -function satisfying the PDE (4). Then, there exist positive constants α and C , depending only on n , such that [figure omitted; refer to PDF]

Proof.

Step 1 . We will prove that there exists a constant C depending only on n such that [figure omitted; refer to PDF]

To this end, consider the function [figure omitted; refer to PDF] defined on _{B a} ( _{p 0} , G ~ ) , where α is a positive constant to be determined later. Obviously, F attains its supremum at some interior point ^{p *} . We may assume that ^{s 2} is a ^{C 2} -function in a neighborhood of ^{p *} . Choose an orthonormal frame field on M around ^{p *} with respect to the Calabi metric G . Then, at ^{p *} , [figure omitted; refer to PDF] where "," denotes the covariant derivative with respect to the Calabi metric G as before, and we used the fact ^{|| ∇ s || G 2} = exp ... { α ( f + u ) } . Inserting Proposition 6 into (35), we get [figure omitted; refer to PDF] Combining (34) with (36) and using the Schwarz inequality, we have [figure omitted; refer to PDF] Choose α small enough such that [figure omitted; refer to PDF] Then, by substituting the three estimates above, we get [figure omitted; refer to PDF] here and later C denotes positive constant depending only on n .

Denote ^{a *} = s ( _{p 0} , ^{p *} ) . If ^{a *} = 0 , from (39), it is easy to complete the proof of the lemma. In the following, we assume that ^{a *} > 0 . Now, we calculate the term 4 s Δ s / ( ^{a 2} - ^{s 2} ) . Firstly, we will give a lower bound of the Ricci curvature Ric ( M , G ~ ) . Assume that [figure omitted; refer to PDF] For any p ∈ _{B ^{a *}} ( _{p 0} , G ~ ) , by a coordinate transformation, _{f i j} ( p ) = _{δ i j} and _{R i j} ( p ) = 0 hold for i ...0; j . Then, at p , [figure omitted; refer to PDF] Then, using the Schwarz inequality and (22)-(24), we know that at the point p [figure omitted; refer to PDF] If 3 ( n - 2 ) α ( f + u ) + 2 ( n - 1 ) ...4; 0 , then [figure omitted; refer to PDF] Otherwise, [figure omitted; refer to PDF] Then, the Ricci curvature Ric ( M , G ~ ) on _{B ^{a *}} ( _{p 0} , G ~ ) is bounded from below by [figure omitted; refer to PDF] By the Laplacian comparison theorem, we get [figure omitted; refer to PDF] where Δ ~ denotes the Laplacian with respect to the metric G ~ .

Substituting (46) into (39) yields [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] Multiplying by ( ^{a 2} - ^{s 2 ) 2} ( ^{p *} ) , at both sides of (47), yields [figure omitted; refer to PDF] Using the Schwarz inequality, we complete Step 1.

Step 2 . We will prove that there is a constant C depending only on n such that [figure omitted; refer to PDF] Consider [figure omitted; refer to PDF] defined on _{B a} ( _{p 0} , G ~ ) , where α is the constant in (38). Obviously, H attains its supremum at some interior point ^{q *} . Choose an orthonormal frame field on M around ^{q *} with respect to the Calabi metric G . Then, at ^{q *} , [figure omitted; refer to PDF] where "," denotes the covariant derivative with respect to the Calabi metric G as before. Inserting Corollary 8 into (53), we get [figure omitted; refer to PDF] Applying the Schwarz inequality, we have [figure omitted; refer to PDF] Inserting these estimates into (54) yields [figure omitted; refer to PDF] here and later C denotes different positive constants depending only on n .

We discuss two subcases.

Case 1 . If [figure omitted; refer to PDF] then B ...4; A . In this case, Step 2 is complete.

Case 2 . Now, assume that [figure omitted; refer to PDF] Then, 1 > ( Φ / J ) ( ^{q *} ) . Thus, [figure omitted; refer to PDF] The rest of the estimate is almost the same as in Step 1. The only difference is to deal with the term ( f + u ) . If ( f + u ) ( ^{q *} ) ...4; 0 , then - C ( f + u ) ( ^{q *} ) ...5; 0 . We can drop this term.

Otherwise, exp ... { - α ( f + u ) } ( f + u ) has a uniform upper bound.

Using the same method as in Step 1, we can estimate the term 4 s Δ s / ( ^{a 2} - ^{s 2} ) and finally get [figure omitted; refer to PDF] Then, combining the conclusion of Step 1, we get [figure omitted; refer to PDF] This completes the proof of Lemma 9.

Proof of Theorem 2.

For any point q ∈ M , choose sufficient large constant _{R 0} such that q ∈ _{B R 0} ( _{p 0} , G ~ ) . Then, for all a ...5; _{R 0} , q ∈ _{B a} ( _{p 0} ) . Using Lemma 9, we know [figure omitted; refer to PDF] Now, let a [arrow right] + ∞ , and we have [figure omitted; refer to PDF] Consequently, [figure omitted; refer to PDF] This completes the proof of Theorem 2.

4. Appendix

Proof of Proposition 6.

Let p ∈ M , and we choose a local orthonormal frame field of the metric G around p . Then, [figure omitted; refer to PDF] where we used (20). In the case Φ ( p ) = 0 , it is easy to get, at p , [figure omitted; refer to PDF] Now, we assume that Φ ( p ) ...0; 0 . Choose a local orthonormal frame field of the metric G around p such that _{ρ , 1} ( p ) = || ∇ ρ || ( p ) > 0 , _{ρ , i} ( p ) = 0 , for all i > 1 . Then, [figure omitted; refer to PDF] where 1 > δ > 0 is a constant to be determined later. Applying (20), we obtain [figure omitted; refer to PDF] An application of the Ricci identity shows that [figure omitted; refer to PDF] Substituting (68) and (69) into (67), we obtain [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] Then, (70) and (71) together give us [figure omitted; refer to PDF] Using the Schwarz inequality gives [figure omitted; refer to PDF] Using [figure omitted; refer to PDF] and choosing δ = 7 / ( 3 n + 4 ) , we get [figure omitted; refer to PDF]

In the following, we will calculate the terms _{R 11} ( ( _{ρ , 1} ^{) 2} / ^{ρ 2} ) and ( _{ρ , 1} / ^{ρ 2} ) ( ρ τ _{) , 1} . Note that (17) is invariant under an affine transformation of coordinates that preserved the origin. So, we can choose the coordinates _{x 1} , _{x 2} , ... , _{x n} such that _{f i j} ( p ) = _{δ i j} and ∂ ρ / ∂ _{x 1} = || grad ρ || ( p ) > 0 , ( ∂ ρ / ∂ _{x i} ) ( p ) = 0 , for all i > 1 . From (19), we easily obtain [figure omitted; refer to PDF] Thus, we get [figure omitted; refer to PDF] By the same method, as deriving (69), we have [figure omitted; refer to PDF] Note that ∑ ... _{A i i 1} = ( ( n + 2 ) / 2 ) ( _{ρ 1} / ρ ) . Therefore, by (14), (77), (78), and (79), we obtain [figure omitted; refer to PDF] On the other hand, we have [figure omitted; refer to PDF] Then, inserting (80) and (81) into (75), we get [figure omitted; refer to PDF] Using (77), we have [figure omitted; refer to PDF] One observes that the Schwarz inequality gives [figure omitted; refer to PDF] Note that by (17) we have [figure omitted; refer to PDF] Then, inserting these estimates into (82) yields Proposition 6.

Proof of Corollary 8.

Now, we will calculate the term ( ln ... ρ _{) , i j k} . In particular, if f satisfies PDE (4), choose the coordinate ( _{x 1} , _{x 2} , ... , _{x n} ) such that _{f i j} ( p ) = _{δ i j} ; then we have [figure omitted; refer to PDF] Using (17), we have [figure omitted; refer to PDF] By the Young inequality and the Schwarz inequality, we have [figure omitted; refer to PDF] Thus, by inserting (88) into Lemma 7, we obtain Corollary 8.

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).

Conflict of Interests

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

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