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

Abstract

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 $\mathrm{\Omega }\subset {\mathbb{R}}^n$; then the graph $M_{\nabla f}$ of $\nabla f$ is a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean space ${\mathbb{R}}_n^{\mathrm{2}n}$ 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 $R_n^{\mathrm{2}n}$ and passes through the origin then it is flat.