Non-preserved curvature conditions under constrained mean curvature flows |
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Institution: | 1. Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, 60054 Frankfurt, Germany;2. Department of Mathematics, University of Valencia, calle Dr. Moliner 50, 46100 Burjassot, Valencia, Spain |
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Abstract: | We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow. |
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Keywords: | Volume preserving mean curvature flow Area preserving mean curvature flow Mean convex hypersurface Scalar curvature |
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