Liouville-type theorems and an application to the mean curvature flow

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Doug Stryker, Princeton University

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The classical Liouville theorem asserts that bounded entire harmonic functions on \R^n are constant. The usual proof by derivative estimates can be used to show more generally that the space of ancient solutions to the heat equation on \R^n with bounded polynomial growth is finite dimensional. We introduce a more flexible approach to these results due to Colding and Minicozzi that carries over easily to many other settings. In particular, we motivate their approach with an application to the mean curvature flow in high codimension.