We prove a rigidity result in the sphere which allows us to generalize a result about smooth convex hypersurfaces in the sphere by Do Carmo-Warner to convex C 2 -hypersurfaces. We apply these results to prove C 1,βconvergence of inverse F -curvature flows in the sphere to an equator in S n+1 for embedded, closed, strictly convex initial hypersurfaces. The result holds for large classes of curvature functions including the mean curvature and arbitrary powers of the Gauss curvature. We use this result to prove Alexandrov-Fenchel type inequalities in the sphere.

We consider inverse curvature flows in hyperbolic space with starshaped initial hypersurface, driven by positive powers of a homogeneous curvature function. The solutions exist for all time and, after rescaling, converge to a sphere.Comment: The rescaling, under which the flow hypersurfaces converge to a constant, had to be modified due to a mistake in the corresponding proof in the first version. Several typing errors were correcte

We study long-time existence and asymptotic behavior for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even Lp-Minkowski problems in R n+1 for p > −n − 1.

We consider the smooth inverse mean curvature flow of strictly convex hypersurfaces with boundary embedded in $\mathbb{R}^{n+1},$ which are perpendicular to the unit sphere from the inside. We prove that the flow hypersurfaces converge to the embedding of a flat disk in the norm of $C^{1,\beta},$ $\beta<1.$Comment: 18 pages. Comments or suggestions are welcom

We consider the inverse mean curvature flow in smooth Riemannian manifolds of the form ([R0, ∞) × S n ,ḡ) with metricḡ = dr 2 + ϑ 2 (r)σ and non-positive radial sectional curvature. We prove, that for initial mean-convex graphs over S n the flow exists for all times and remains a graph over S n . Under weak further assumptions on the ambient manifold, we prove optimal decay of the gradient and that the flow leaves become umbilic exponentially fast. We prove optimal C 2estimates in case that the ambient pinching improves.