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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 $k \geq 1$ 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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