On inverse mean curvature flow in Schwarzschild space and Kottler space

Abstract

In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface $\Sigma$ is strictly mean convex and star-shaped, then the flow hypersurface ${\Sigma }_t$ converges to a large coordinate sphere as $t\to \infty$ exponentially. We also describe an application of this convergence result. In the second part of this paper, we will analyse the inverse mean curvature flow in Kottler-Schwarzchild manifold. By deriving a lower bound for the mean curvature on the flow hypersurface independently of the initial mean curvature, we can use an approximation argument to show the global existence and regularity of the smooth inverse mean curvature flow for star-shaped and weakly mean convex initial hypersurface, which generalizes Huisken-Ilmanen's result [18].

23 pages, v2, title changed, new result added

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Cited by in corpus (5)

• On the Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space
• Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature
• Inverse curvature flows in Riemannian warped products
• Inverse mean curvature flows in warped product manifolds
• A Penrose-type inequality for static spacetimes