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The topology of hypersurfaces moving by mean curvature. (English) Zbl 0858.58047
The levels \(M_t\) of the mean curvature flow may become singular as \(t\to T\). Beyond the parameter \(T\) there are still weak solutions, the so-called weak mean curvature flow. This paper addresses the question: what topological changes of \(M_t\) are possible? Roughly, the main result states that two distinct connected components of the complement \(W [0]\) of \(M_0\) cannot become connected together later, and that connected components cannot appear out of nowhere. This is expressed by an induced isomorphism in the homology \(H_0 (W [0]) \to H_0 (W[0,T])\). Similarly, every loop in \(W[0,T]\) is homotopic to a loop in \(W[0]\). Furthermore, if the ambient space is an \((n+1)\)-dimensional Riemannian manifold with Ricci curvature bounded below, then there is a similar result for the homologies \(H_n\) and \(H_{n-1}\). A number of examples illustrate various topology changes which actually can occur after singularities of the flow.

MSC:
58J35 Heat and other parabolic equation methods for PDEs on manifolds
37C10 Dynamics induced by flows and semiflows
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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