On the rigidity theorems for Lagrangian translating solitons in pseudo-Euclidean space II

Abstract

Let $u$ be a smooth convex function in ${\mathbb{R}}^n$ and the graph $M_{\nabla u}$ of $\nabla u$ be a space-like translating soliton in pseudo-Euclidean space ${\mathbb{R}}_n^{2n}$ with a translating vector $\frac{1}{n}(a_1,a_2,\cdots ,a_n;b_1,b_2,\cdots ,b_n)$, then the function $u$ satisfies $$\det D^{2}u=\exp \left\{\sum _{i=1}^n-a_i\frac{\partial u}{\partial x_i}+\sum _{i=1}^n{b}_i{x}_i+c\right\}on{\mathbb{R}}^n$$ where $a_i$, $b_i$ and $c$ are constants. The Bernstein type results are obtained in the course of the arguments.

9 pages

References in corpus (3)

• The Clifford torus as a self-shrinker for the Lagrangian mean curvature flow
• On the self-shrinking systems in arbitrary codimension spaces
• On the topology of translating solitons of the mean curvature flow