Complete self-shrinking solutions for Lagrangian mean curvature flow in pseudo-Euclidean space. (English) Zbl 1476.53114

Xu, Ruiwei; Cao, Linfen

Abstr. Appl. Anal. 2014, Article ID 196751, 9 p. (2014).

Summary: Let \(f(x)\) be a smooth strictly convex solution of \(\text{det}(\partial^2 f / \partial x_i \partial x_j) = \text{exp} \left\{(1 / 2) \sum_{i = 1}^n x_i(\partial f / \partial x_i) - f\right\}\) defined on a domain \(\Omega \subset \mathbb{R}^n\); then the graph \(M_{\nabla f}\) of \(\nabla f\) is a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean space \(\mathbb{R}_n^{2 n}\) with the indefinite metric \(\sum d x_i d y_i\). In this paper, we prove a Bernstein theorem for complete self-shrinkers. As a corollary, we obtain if the Lagrangian graph \(M_{\nabla f}\) is complete in \(R_n^{2 n}\) and passes through the origin then it is flat.

MSC:

53E10	Flows related to mean curvature
53A35	Non-Euclidean differential geometry
35K55	Nonlinear parabolic equations
53E50	Flows related to symplectic and contact structures

Keywords:

space-like self-shrinker; pseudo-Euclidean space; complete self-shrinkers; Lagrangian graph

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