Complete Self-Shrinking Solutions for Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space - Ruiwei Xu; Linfen Cao (Abstract and Applied Analysis)
Let f(x) be a smooth strictly convex solution of det(∂2f/∂xi∂xj)=exp(1/2)∑i=1nxi(∂f/∂xi)-f defined on a domain Ω⊂Rn; then the graph M∇f of ∇f is a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean space Rn2n with the indefinite metric ∑dxidyi. 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 Rn2n and passes through the origin then it is flat.