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Contracting Convex Hypersurfaces in Space Form by Non-homogeneous Curvature Function

  • Guanghan Li
  • Yusha LvEmail author
Article
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Abstract

This paper concerns the evolution of closed convex hypersurfaces in space form \({\mathbb {N}}^{n+1}_\kappa \) of constant sectional curvature \(\kappa =0, \pm 1\) by curvature flow, for which the normal speed is given by a general non-homogeneous function \(\varPhi \) of curvature function F which is monotone, symmetric, homogeneous of degree one. Under the assumption that F is inverse concave and its dual function \(F_*\) approaches zero on the boundary of positive cone, we prove that the flow converges to a point in finite time. For special flow with \(\varPhi =F^\beta ~(\beta >1)\), without the assumption of inverse concavity on F, we deduce that if the initial hypersurface is pinched enough, then the flow converges smoothly and exponentially to the unit sphere of \({\mathbb {R}}^{n+1}\) after some suitable rescaling.

Keywords

Non-homogeneous curvature flow Convex hypersurfaces Space form 

Mathematics Subject Classification

53C44 35K55 
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Notes

Acknowledgements

The authors would like to thank the referees for careful reading of the manuscript and their valuable comments.

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Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

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