Entire mean curvature flows of graphs

Abstract

We consider the evolution of the graph of f : ‫ޒ‬ n → ‫ޒ‬ n in ‫ޒ‬ n × ‫ޒ‬ n by the mean curvature flow. We prove that the flow exists smoothly for all time if the differential of f has a positive lower bound. Moreover, at each time, the flow remains the graph of a map f t .