Asymptotic Convergence for a Class of Fully Nonlinear Curvature Flows

Qi Rui Li*, Weimin Sheng, Xu Jia Wang

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    10 Citations (Scopus)

    Abstract

    In this paper, we study a class of contracting flows of closed, convex hypersurfaces in the Euclidean space Rn + 1 with speed rασk, where σk is the k-th elementary symmetric polynomial of the principal curvatures, α∈ R1, and r is the distance from the hypersurface to the origin. If α≥ k+ 1 , we prove that the flow exists for all time, preserves the convexity and converges smoothly after normalisation to a sphere centred at the origin. If α< k+ 1 , a counterexample is given for the above convergence. In the case k= 1 and α≥ 2 , we also prove that the flow converges to a round point if the initial hypersurface is weakly mean-convex and star-shaped.

    Original languageEnglish
    Pages (from-to)834-860
    Number of pages27
    JournalJournal of Geometric Analysis
    Volume30
    Issue number1
    DOIs
    Publication statusPublished - 1 Jan 2020

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