Flow by powers of the Gauss curvature

Ben Andrews; Pengfei Guan; Lei Ni

doi:10.1016/j.aim.2016.05.008

Flow by powers of the Gauss curvature

Ben Andrews, Pengfei Guan^*, Lei Ni

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

62 Citations (SciVal)

Abstract

We prove that convex hypersurfaces in Rⁿ⁺¹ contracting under the flow by any power α>1/n+2 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the initial body we prove that the limit is the round sphere for α≥1.

Original language	English
Pages (from-to)	174-201
Number of pages	28
Journal	Advances in Mathematics
Volume	299
DOIs	https://doi.org/10.1016/j.aim.2016.05.008
Publication status	Published - 20 Aug 2016

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title = "Flow by powers of the Gauss curvature",

abstract = "We prove that convex hypersurfaces in Rn+1 contracting under the flow by any power α>1/n+2 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the initial body we prove that the limit is the round sphere for α≥1.",

keywords = "Curvature Image, Entropy, Entropy Stability Estimates, Gauss Curvature, Monotonicity, Regularity Estimates",

author = "Ben Andrews and Pengfei Guan and Lei Ni",

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AU - Ni, Lei

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N2 - We prove that convex hypersurfaces in Rn+1 contracting under the flow by any power α>1/n+2 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the initial body we prove that the limit is the round sphere for α≥1.

AB - We prove that convex hypersurfaces in Rn+1 contracting under the flow by any power α>1/n+2 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the initial body we prove that the limit is the round sphere for α≥1.

KW - Curvature Image

KW - Entropy

KW - Entropy Stability Estimates

KW - Gauss Curvature

KW - Monotonicity

KW - Regularity Estimates

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DO - 10.1016/j.aim.2016.05.008

M3 - Article

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VL - 299

SP - 174

EP - 201

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

Flow by powers of the Gauss curvature

Abstract

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