Slow convergence of graphs under mean curvature flow

In this paper, we study the mean curvature flow of entire graphs in Euclidean space. Ecker and Huisken have shown that given some initial growth condition at infinity and bounded initial gradient, such graphs, when rescaled, become self-similar under this evolution. Furthermore the convergence is exponentially fast in time. Here we consider a weaker condition at infinity, and show that under mean curvature flow such a condition is preserved for the height of the graph during the extent of the evolution. Our main result then states that under this alternative condition at infinity and bounded gradient, the rescaled graphs also become self-similar, converging however at a slower (polynomial in time) rate.

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Published 1 January 2010