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

and Applied Analysis 3


Introduction
Let  be an -dimensional submanifold immersed into the Euclidean space R + .Mean curvature flow is a oneparameter family   = (⋅, ) of immersions   :  → R + with corresponding images   =   () such that    (, ) =  (, ) ,  ∈ ,  (, 0) =  () is satisfied, where (, ) is the mean curvature vector of   at (, ) in R + .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][2][3][4][5][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][9][10][11][12][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.
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  2  , then is it flat?In this paper, we will use affine technique (see [15][16][17][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 Ω ⊆ R  satisfying the PDE (4).If there is a positive constant  depending only on  such that the hypersurface  = {(, ())} in R +1 is complete with respect to the metric then  is the quadratic polynomial.
As a direct application of Theorem 2, we have the following.

Corollary 4. Let 𝑓 be a strictly convex 𝐶
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 𝑀
is a complete space-like self-shrinker for mean curvature flow and passes through the origin, then  ∇ is flat.
For , we choose the canonical relative normalization  = (0, 0, . . ., 1).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  = ( 1 , . . .,   , ( 1 , . . .,   )), 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  := (1/2) ∑   (  /)  .From ( 19), we have By ( 17), we get and then Using (17) yields Define a conformal Riemannian metric G := exp{( + )}, where  is a constant.
Conformal Ricci Curvature.Denote by R the Ricci curvature with respect to the metric G; 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 ( 1 , . . .,   ) 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 𝐽 satisfies
where ", " denotes the covariant derivative with respect to the Calabi metric.

Proof of Theorem 2
It is our aim to prove Φ ≡ 0; thus, from definition of , everywhere on .As in [8], by Euler homogeneous theorem, we get Theorem 2.
Step 1.We will prove that there exists a constant  depending only on  such that To this end, consider the function defined on   ( 0 , G), where  is a positive constant to be determined later.Obviously,  attains its supremum at some interior point  * .We may assume that  2 is a  2 -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 ‖∇‖ 2  = exp{( + )}.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 .
Step 2. We will prove that there is a constant  depending only on  such that 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.
then B ≤ A. In this case, Step 2 is complete.
Using the same method as in Step 1, we can estimate the term 4Δ/( 2 −  2 ) and finally get Then, combining the conclusion of Step 1, we get This completes the proof of Lemma 9.