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The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1555–1570 | Cite as

New Pinching Estimates for Inverse Curvature Flows in Space Forms

  • Yong WeiEmail author
Article

Abstract

We consider the inverse curvature flow of strictly convex hypersurfaces in the space form N of constant sectional curvature \(K_N\) with speed given by \(F^{-\alpha }\), where \(\alpha \in (0,1]\) for \(K_N=0,-1\) and \(\alpha =1\) for \(K_N=1\), F is a smooth, symmetric homogeneous of degree one function which is inverse concave and has dual \(F_*\) approaching zero on the boundary of the positive cone \(\Gamma _+\). We show that the ratio of the largest principal curvature to the smallest principal curvature of the flow hypersurface is controlled by its initial value. This can be used to prove the smooth convergence of the flows.

Keywords

Pinching estimate Inverse curvature flow Space form Inverse concave 

Mathematics Subject Classification

53C44 53C21 

Notes

Acknowledgements

The author would like to thank the referees for carefully reading of this manuscript and providing many helpful suggestions. The author was supported by Ben Andrews throughout his Australian Laureate Fellowship FL150100126 of the Australian Research Council.

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia

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