GENERALIZED FLOW OF SETS BY MEAN-CURVATURE ON A MANIFOLD

The level-set flow of Evans-Spruck and Chen-Giga-Goto is generalized to a Riemannian manifold, using recent techniques of Crandall-Ishii 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 C2.