Inverse mean curvature flows in warped product manifolds

Abstract

We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor $\phi (r)$. If ${\phi }^{\prime }(r)>0$ and ${\phi }^{\prime \prime }(r)\ge 0$, we show that these flows exist for all times, remain starshaped and mean convex. Plus the positivity of ${\phi }^{\prime \prime }(r)$ and a curvature condition we obtain a lower positive bound of mean curvature along these flows independent of the initial mean curvature. We also give a sufficient condition to extend the asymptotic behavior of these flows in Euclidean spaces into some more general warped product manifolds.

Version 4: Final Version. Update some remarks for the conditions(see Remark 5.4). To appear the Journal of Geometric Analysis. Version 5: correct a reference typo and add two new references, mention an unpublished work in this direction

References in corpus (1)

• Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold

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