The rigidity theorems for Lagrangian self shrinkers

Abstract

By the integral method we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in ${\mathbb{R}}_n^{2n}$ with the indefinite metric ${\sum }_{i}d{x}_{i}d{y}_i$ is flat. This result improves the previous ones in \cite{HW} and \cite{CCY} by removing the additional assumption in their results. In a similar manner, we reprove its Euclidean counterpart which is established in \cite{CCY}.

16 pages