Convex ancient solutions to mean curvature flow

Theodora Bourni; Mat Langford; Giuseppe Tinaglia

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arXiv · 1907.03932

Convex ancient solutions to mean curvature flow

Theodora Bourni·Mat Langford·Giuseppe Tinaglia

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Abstract

X.-J. Wang proved a series of remarkable results on the structure of convex ancient solutions to mean curvature flow. Some of his results do not appear to be widely known, however, possibly due to the technical nature of his arguments and his exploitation of methods which are not widely used in mean curvature flow. In this expository article, we present Wang's structure theory and some of its consequences. We shall simplify some of Wang's analysis by making use of the monotonicity formula and the differential Harnack inequality, and obtain an important additional structure result by exploiting the latter. We conclude by showing that various rigidity results for convex ancient solutions and convex translators follow quite directly from the structure theory, including the new result of Corollary 8.3}. We recently provided a complete classification of convex ancient solutions to curve shortening flow by exploiting similar arguments.

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Theodora Bourni, Mat Langford, Giuseppe Tinaglia. 2019-07-09. Convex ancient solutions to mean curvature flow. https://arxiv.org/abs/1907.03932

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