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

Let f(x) be a smooth strictly convex solution of det(∂² f /∂x_i∂x_i) = exp {(1/2) Σ_i=1ⁿ x_i (∂f /∂x_i) - 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 ℝ_n²ⁿ with the indefinite metric Σdx_idy_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_n²ⁿ and passes through the origin then it is flat. [ABSTRACT FROM AUTHOR]