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Generalized flow of sets by mean curvature on a manifold. (English) Zbl 0759.53035
The level-set flow of L. Evans and J. Spruck [part I: J. Differ. Geom. 33, 635-681 (1991; Zbl 0726.53029), parts II-IV to appear] and Y.-G. Chen, Y. Giga and S. Goto [J. Differ. Geom. 33, 749-786 (1991; Zbl 0696.35087)] is generalized to a Riemannian manifold, using recent techniques of M. Crandall and H. Ichii [Differ. Integral Eq. 3, 1001-1014 (1990; Zbl 0723.35015)] for viscosity solutions. Generally speaking, the motion is not unique for noncompact closed sets, but the definition can be modified to make the motion unique. We give examples to show: 1) a smooth set can develop an interior that originates from infinity; 2) in the case of a Grayson neckpinch, the evolving function \(u(x,t)\) need not remain \(C^ 2\).

MSC:
53C40 Global submanifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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