Non-scale-invariant inverse curvature flows in Euclidean space

Abstract

We consider the inverse curvature flows $\dot x=F^{-p}\nu$ of closed star-shaped hypersurfaces in Euclidean space in case $0<p\not=1$ and prove that the flow exists for all time and converges to infinity, if $0<p<1$, while in case $p>1$, the flow blows up in finite time, and where we assume the initial hypersurface to be strictly convex. In both cases the properly rescaled flows converge to the unit sphere.