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Convexity estimates for flows by powers of the mean curvature

Felix Schulze — 2006

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study the evolution of a closed, convex hypersurface in ℝ n + 1 in direction of its normal vector, where the speed equals a power k ≥ 1 of the mean curvature. We show that if initially the ratio of the biggest and smallest principal curvatures at every point is close enough to 1 , depending only on k and n , then this is maintained under the flow. As a consequence we obtain that, when rescaling appropriately as the flow contracts to a point, the evolving surfaces converge to the unit sphere.

Self-similarly expanding networks to curve shortening flow

Oliver C. Schnürer;  Felix Schulze — 2007

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point, where they form 120 degree angles, and expands homothetically under curve shortening flow. We also prove uniqueness of these networks.

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