The harmonic mean curvature flow of nonconvex surfaces in $\mathbb{R}^3$

Details The harmonic mean curvature flow of nonconvex surfaces in $\mathbb{R}^3$ The harmonic mean curvature flow of nonconvex surfaces in $\mathbb{R}^3$

2008-06-10

arxiv.org/abs/0806.1758v2

Mathematics

Differential Geometry

Scientific paper

We consider a compact, star-shaped, mean convex hypersurface $\Sigma^2\subset \mathbb{R}^3$. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \cite{An1}). We also prove that in the case we have a surface of revolution which is star-shaped and mean convex, a smooth solution always exists up to some finite time $T < \infty$ at which the flow shrinks to a point asymptotically spherically.

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