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February 2021 Volume preserving flow by powers of the $k$th mean curvature
Ben Andrews, Yong Wei
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J. Differential Geom. 117(2): 193-222 (February 2021). DOI: 10.4310/jdg/1612975015

Abstract

We consider the flow of closed convex hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ with speed given by a power of the $k$‑th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume $V_{n+1}$ and the mixed volume $V_{n+1-k}$ of the evolving hypersurface. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to a round sphere smoothly. No curvature pinching assumption is required on the initial hypersurface.

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Ben Andrews. Yong Wei. "Volume preserving flow by powers of the $k$th mean curvature." J. Differential Geom. 117 (2) 193 - 222, February 2021. https://doi.org/10.4310/jdg/1612975015

Information

Received: 29 November 2017; Published: February 2021
First available in Project Euclid: 10 February 2021

Digital Object Identifier: 10.4310/jdg/1612975015

Rights: Copyright © 2021 Lehigh University

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Vol.117 • No. 2 • February 2021
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