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Volume preserving flow by powers of symmetric polynomials in the principal curvatures

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Abstract

We study a volume preserving curvature flow of convex hypersurfaces, driven by a power of the k-th elementary symmetric polynomial in the principal curvatures. Unlike most of the previous works on related problems, we do not require assumptions on the curvature pinching of the initial datum. We prove that the solution exists for all times and that the speed remains bounded and converges to a constant in an integral norm. In the case of the volume preserving scalar curvature flow, we can prove that the hypersurfaces converge smoothly to a round sphere.

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Acknowledgements

Carlo Sinestrari was partially supported by the research group GNAMPA of INdAM (Istituto Nazionale di Alta Matematica).

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Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università di Roma “Roma Tre”, Largo San Leonardo Murialdo 1, 00146, Rome, Italy

    Maria Chiara Bertini

  2. Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma “Tor Vergata”, Via Politecnico 1, 00133, Rome, Italy

    Carlo Sinestrari

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  1. Maria Chiara Bertini
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  2. Carlo Sinestrari
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Correspondence to Carlo Sinestrari.

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Bertini, M.C., Sinestrari, C. Volume preserving flow by powers of symmetric polynomials in the principal curvatures. Math. Z. 289, 1219–1236 (2018). https://doi.org/10.1007/s00209-017-1995-8

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  • DOI: https://doi.org/10.1007/s00209-017-1995-8

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