Non-preserved curvature conditions under constrained mean curvature flows

2014/12/28 by Esther Cabezas-Rivas, Cabezas-Rivas, Esther, Vicente Miquel +1
Mathematics · #53C44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:53C44

paper · pdf · doi:10.48550/arxiv.1412.8143

16 pages, 3 figures

arxiv created 2014/12/28 · arxiv updated 2014/12/30

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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