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Inverse Curvature Flows of Rotation Hypersurfaces

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Abstract

We consider the inverse curvature flows of smooth, closed and strictly convex rotation hypersurfaces in space forms \(\mathbb{M}_{k}^{n+1}\) with speed function given by F−α, where α ∈ (0, 1] for κ = 0, −1, α =1 for κ = 1 and F is a smooth, symmetric, strictly increasing and 1-homogeneous function of the principal curvatures of the evolving hypersurfaces. We show that the curvature pinching ratio of the evolving hypersurface is controlled by its initial value, and prove the long time existence and convergence of the flows. No second derivatives conditions are required on F.

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Acknowledgements

The authors would like to thank the referee for his/her helpful commments.

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, P. R. China

    Yu Han Jin & Xian Feng Wang

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P. R. China

    Yong Wei

Authors
  1. Yu Han Jin
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  2. Xian Feng Wang
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  3. Yong Wei
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Correspondence to Yong Wei.

Additional information

Supported by the National Key R and D Program of China (Grant No. 2020YFA0713100), National Natural Science Foundation of China (Grant Nos. 11971244 and 11871283), Natural Science Foundation of Tianjin, China (Grant No. 19JCQNJC14300), and Research (Grant No. KY0010000052) from University of Science and Technology of China

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Jin, Y.H., Wang, X.F. & Wei, Y. Inverse Curvature Flows of Rotation Hypersurfaces. Acta. Math. Sin.-English Ser. 37, 1692–1708 (2021). https://doi.org/10.1007/s10114-021-0015-4

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  • DOI: https://doi.org/10.1007/s10114-021-0015-4

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