Research Article | Open Access

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

**Academic Editor:**Ljubomir B. Ćirić

#### 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).

#### References

- T. H. Colding and I. Minicozzi, “Generic mean curvature flow I: generic singularities,”
*Annals of Mathematics*, vol. 175, no. 2, pp. 755–833, 2012. View at: Publisher Site | Google Scholar | MathSciNet - K. Ecker and G. Huisken, “Mean curvature evolution of entire graphs,”
*Annals of Mathematics*, vol. 130, no. 3, pp. 453–471, 1989. View at: Publisher Site | Google Scholar | MathSciNet - G. Huisken, “Asymptotic behavior for singularities of the mean curvature flow,”
*Journal of Differential Geometry*, vol. 31, no. 1, pp. 285–299, 1990. View at: Google Scholar | MathSciNet - G. Huisken, “Local and global behaviour of hypersurfaces moving by mean curvature,” in
*Differential Geometry: Partial Differential Equations on Manifolds*, vol. 54 of*Proceedings of Symposia in Pure Mathematics*, American Mathematical Society, 1993. View at: Google Scholar - K. Smoczyk, “Self-shrinkers of the mean curvature flow in arbitrary codimension,”
*International Mathematics Research Notices*, no. 48, pp. 2983–3004, 2005. View at: Publisher Site | Google Scholar | MathSciNet - L. Wang, “A Bernstein type theorem for self-similar shrinkers,”
*Geometriae Dedicata*, vol. 151, pp. 297–303, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. D. Cao and H. Li, “A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension,”
*Calculus of Variations and Partial Differential Equations*, vol. 46, no. 3-4, pp. 879–889, 2013. View at: Publisher Site | Google Scholar | MathSciNet - A. Chau, J. Chen, and Y. Yuan, “Rigidity of entire self-shrinking solutions to curvature flows,”
*Journal für die Reine und Angewandte Mathematik*, vol. 664, pp. 229–239, 2012. View at: Publisher Site | Google Scholar | MathSciNet - Q. Ding and Z. Z. Wang,
*On the self-shrinking systems in arbitrary codimension spaces*, http://arxiv.org/abs/1012.0429. - Q. Ding and Y. L. Xin, “The Rigidity theorems for Lagrangian self-shrinkers,”
*Journal für die Reine und Angewandte Mathematik*, 2012. View at: Publisher Site | Google Scholar - K. Ecker, “Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space,”
*Journal of Differential Geometry*, vol. 46, no. 3, pp. 481–498, 1997. View at: Google Scholar | Zentralblatt MATH | MathSciNet - R. Huang and Z. Wang, “On the entire self-shrinking solutions to Lagrangian mean curvature flow,”
*Calculus of Variations and Partial Differential Equations*, vol. 41, no. 3-4, pp. 321–339, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. L. Xin, “Mean curvature flow with bounded Gauss image,”
*Results in Mathematics*, vol. 59, no. 3-4, pp. 415–436, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. L. Xin,
*Minimal Submanifolds and Related Topics*, World Scientific Publishing, 2003. View at: MathSciNet - A.-M. Li, R. Xu, U. Simon, and F. Jia,
*Affine Bernstein Problems and Monge-Ampère Equations*, World Scientific, Singapore, 2010. View at: MathSciNet - A. M. Li and R. W. Xu, “A rigidity theorem for an affine Kähler-Ricci flat graph,”
*Results in Mathematics*, vol. 56, no. 1–4, pp. 141–164, 2009. View at: Publisher Site | Google Scholar | MathSciNet - A.-M. Li and R. Xu, “A cubic form differential inequality with applications to affine Kähler-Ricci flat manifolds,”
*Results in Mathematics*, vol. 54, no. 3-4, pp. 329–340, 2009. View at: Publisher Site | Google Scholar | MathSciNet - R. W. Xu and R. L. Huang, “On the rigidity theorems for Lagrangian translating solitons in pseudo-Euclidean space I,”
*Acta Mathematica Sinica (English Series)*, vol. 29, no. 7, pp. 1369–1380, 2013. View at: Publisher Site | Google Scholar | MathSciNet - A. V. Pogorelov,
*The Minkowski Multidimensional Problem*, John Wiley & Sons, New York, NY, USA, 1978. - E. Calabi, “Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens,”
*The Michigan Mathematical Journal*, vol. 5, pp. 105–126, 1958. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Ruiwei Xu and Linfen Cao. 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.