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arXiv · 2309.16449

Mean curvature flows of graphs sliding off to infinity in warped product manifolds

Naotoshi Fujihara
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Abstract

We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and $\mathbb{R}$. In such a warped product manifold, we can define the notion of a graph, called a geodesic graph. We prove that the curve shortening flow preserves a geodesic graph for any warping function, and the mean curvature flow of hypersurfaces preserves a geodesic graph for some monotone convex warping functions. In particular, we consider some warping functions that go to zero at infinity, which means that the curves or hypersurfaces go to a point at infinity along the flow. In such a case, we prove the long-time existence of the flow and that the curvature and its higher-order derivatives go to zero along the flow.

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Naotoshi Fujihara. 2023-09-28. Mean curvature flows of graphs sliding off to infinity in warped product manifolds. https://arxiv.org/abs/2309.16449

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