Paper · May 25, 2018

Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights

Ana Hurtado, Vicente Palmer, César Rosales

math.DGWider flowsarXiv:1805.10055
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Abstract

Let be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight . The aim of this paper is twofold. First, by assuming certain control on the -mean curvature of , we establish comparisons for the -capacity of extrinsic balls in , from which we deduce criteria ensuring the -parabolicity or -hyperbolicity of . Second, we employ functions with geometric meaning to describe submanifolds of bounded -mean curvature which are confined into some regions of the ambient manifold. As a consequence, we derive half-space and Bernstein-type theorems generalizing previous ones. Our results apply for some relevant -minimal submanifolds appearing in the singularity theory of the mean curvature flow.

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Submitted
May 25, 2018
Subjects
math.DG, math.AP

Comments: 32 pages, no figures

Mean curvature flowSingularitiesRotational symmetry & warped productsMinimal & CMC surfaces

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  6. Analysis of Velzquez's solution to the mean curvature flow with a type singularity

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  7. Mean curvature flow in Fuchsian manifolds

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  8. A strong stability condition on minimal submanifolds and its implications

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

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