# zbMATH — the first resource for mathematics

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.)
##### Keywords:
maximum principle; level set flow; mean curvature flow
Full Text: