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Annals of Global Analysis and Geometry

, Volume 55, Issue 2, pp 243–279 | Cite as

Surfaces expanding by non-concave curvature functions

  • Haizhong Li
  • Xianfeng WangEmail author
  • Yong Wei
Article

Abstract

In this paper, we first investigate the flow of convex surfaces in the space form \(\mathbb {R}^3(\kappa )~(\kappa =0,1,-1)\) expanding by \(F^{-\alpha }\), where F is a smooth, symmetric, increasing and homogeneous of degree one function of the principal curvatures of the surfaces and the power \(\alpha \in (0,1]\) for \(\kappa =0,-1\) and \(\alpha =1\) for \(\kappa =1\). By deriving that the pinching ratio of the flow surface \(M_t\) is no greater than that of the initial surface \(M_0\), we prove the long time existence and the convergence of the flow. No concavity assumption of F is required. We also show that for the flow in \(\mathbb {H}^3\) with \(\alpha \in (0,1)\), the limit shape may not be necessarily round after rescaling.

Keywords

Surface Space form Inverse curvature flow Non-concave curvature function 

Mathematics Subject Classification

Primary 53C44 Secondary 53C21 58J35 

Notes

Acknowledgements

The first author was supported in part by NSFC Grant No. 11671214. The second author was supported in part by NSFC Grant No. 11571185 and the Fundamental Research Funds for the Central Universities. The third author was supported by Ben Andrews throughout his Australian Laureate Fellowship FL150100126 of the Australian Research Council. The authors would like to thank Ben Andrews and Julian Scheuer for comments on the earlier version of this paper and the referees for carefully reading this paper and providing many helpful suggestions.

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China
  2. 2.School of Mathematical Sciences and LPMCNankai UniversityTianjinPeople’s Republic of China
  3. 3.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia

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