Title: Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature,
Abstract: We prove convergence results for expanding curvature flows in the Euclidean
and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and
$F$ is a positive, strictly monotone and 1-homogeneous curvature function. In
particular this class includes the mean curvature $F=H$. We prove that a
certain initial pinching condition is preserved and the properly rescaled
hypersurfaces converge smoothly to the unit sphere. We show that an example due
to Andrews-McCoy-Zheng can be used to construct strictly convex initial
hypersurfaces, for which the inverse mean curvature flow to the power $p>1$
loses convexity, justifying the necessity to impose a certain pinching
condition on the initial hypersurface.