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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.

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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