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

TL;DR

This paper proves rigidity theorems for Lagrangian translating solitons in pseudo-Euclidean space, showing that certain convex solutions must be quadratic, extending Bernstein-type results in this geometric context.

Contribution

It establishes new rigidity results for Lagrangian translating solitons in pseudo-Euclidean space, characterizing solutions as quadratic functions under specific conditions.

Findings

Solutions are necessarily quadratic under given conditions.

Derived a specific Monge-Ampère type equation for convex functions.

Extended Bernstein-type theorems to pseudo-Euclidean space.

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 $\mathbb{R}^{2n}_{n}$ 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 $$ \det D^{2}u=\exp \left\{ \sum_{i=1}^n- a_i\frac{\partial u}{\partial x_{i}} +\sum_{i=1}^n b_ix_i+c\right\} \qquad \hbox{on}\qquad\mathbb R^n$$ where $a_i$, $b_i$ and $c$ are constants. The Bernstein type results are obtained in the course of the arguments.