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More Mixed Volume Preserving Curvature Flows

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Abstract

We extend the results of McCoy (Calc Var Partial Differ Equ 24:131–154, 2005) to include several new cases where convex surfaces evolve to spheres under mixed volume preserving curvature flows, using recent results for unconstrained curvature flows and new regularity arguments in the constrained flow setting. We include results for speeds that are degree 1 homogeneous in the principal curvatures and indicate how, with sufficient curvature pinching conditions on the initial hypersurfaces, some results may be extended to speed homogeneous of degree \(\alpha >1\). In particular, these extensions require lower speed bounds that are obtained here without using estimates for equations of porous medium type, in contrast to most previous work.

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Acknowledgements

This work was completed while the author was supported by DP150100375 of the Australian Research Council. The author is grateful to the Faculty of Engineering and Information Sciences at the University of Wollongong for the provision of a period of study leave that has also assisted in facilitating the completion of this article. Some of this work was carried out while the author was visiting the Centre for Mathematics and its Applications at the Australian National University; the author is grateful for its hospitality and useful discussions with Professor Ben Andrews, particularly his suggestions for the cases of higher homogeneity speeds. The author would also like to thank Professor Neil Trudinger for his interest in this work, Dr Glen Wheeler and Dr Valentina Wheeler for useful discussions.

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  1. Institute for Mathematics and Its Applications, University of Wollongong, Northfields Av, Wollongong, NSW, 2522, Australia

    James A. McCoy

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  1. James A. McCoy
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McCoy, J.A. More Mixed Volume Preserving Curvature Flows. J Geom Anal 27, 3140–3165 (2017). https://doi.org/10.1007/s12220-017-9799-y

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