Motion of hypersurfaces by Gauss curvature

Ben Andrews*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    108 Citations (Scopus)

    Abstract

    We consider n-dimensional convex Euclidean hypersurfaces moving with normal velocity proportional to a positive power α of the Gauss curvature. We prove that hypersurfaces contract to points in finite time, and for α ∈ (1/(n + 2], 1/n] we also prove that in the limit the solutions evolve purely by homothetic contraction to the final point. We prove existence and uniqueness of solutions for non-smooth initial hypersurfaces, and develop upper and lower bounds on the speed and the curvature independent of initial conditions. Applications are given to the flow by affine normal and to the existence of non-spherical homothetically contracting solutions.

    Original languageEnglish
    Pages (from-to)1-34
    Number of pages34
    JournalPacific Journal of Mathematics
    Volume195
    Issue number1
    DOIs
    Publication statusPublished - Sept 2000

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