On the 1/H-flow by p-Laplace approximation: New estimates via fake distances under Ricci lower bounds
In this paper we show the existence of weak solutions $w:M\rightarrow\Bbb{R}$ of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds. Under natural assumptions, we obtain sharp estimates for the growth...
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Published in | American journal of mathematics Vol. 144; no. 3; pp. 779 - 849 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Baltimore
Johns Hopkins University Press
01.06.2022
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Subjects | |
Online Access | Get full text |
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Summary: | In this paper we show the existence of weak solutions $w:M\rightarrow\Bbb{R}$ of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds. Under natural assumptions, we obtain sharp estimates for the growth of $w$ and for the mean curvature of its level sets, which are well behaved with respect to Gromov-Hausdorff convergence. The construction follows R. Moser's approximation procedure via the $p$-Laplace equation, and relies on new gradient and decay estimates for $p$-harmonic capacity potentials, notably for the kernel $\scr{G}_p$ of $\Delta_p$. These bounds, stable as $p\rightarrow 1$, are achieved by studying fake distances associated to capacity potentials and Green kernels. We conclude by investigating some basic isoperimetric properties of the level sets of $w$. |
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ISSN: | 0002-9327 1080-6377 1080-6377 |
DOI: | 10.1353/ajm.2022.0016 |