Complete Self-Shrinking Solutions for Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space. (6th July 2014) - Xu, Ruiwei; Cao, Linfen
Abstract : Letf ( x ) be a smooth strictly convex solution ofdet ( ∂ 2 f / ∂ x i ∂ x j ) = exp ( 1 / 2 ) ∑ i = 1 n x i ( ∂ f / ∂ x i ) - f defined on a domainΩ ⊂ R n ; then the graphM ∇ f of∇ f is a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean spaceR 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 graphM ∇ f is complete inR n 2 n and passes through the origin then it is flat.