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

Abstract EN

Let f ( x ) be a smooth strictly convex solution of det ( ∂ 2 f / ∂ x i ∂ x j ) = exp ( 1 / 2 ) ∑ i = 1 n x i ( ∂ f / ∂ x i ) - f defined on a domain Ω ⊂ R n ; then the graph M ∇ f of ∇ f is a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean space R n 2 n with the indefinite metric ∑ 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 ∇ f is complete in R n 2 n and passes through the origin then it is flat.