Contracting Self-similar Solutions of Nonhomogeneous Curvature Flows
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Contracting Self-similar Solutions of Nonhomogeneous Curvature Flows James A. McCoy1 Received: 19 May 2020 / Accepted: 7 October 2020 © Mathematica Josephina, Inc. 2020
Abstract A recent article (Li and Lv, J Geom Anal 30:417–447, 2020) considered fully nonlinear contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article we consider self-similar solutions to these and related curvature flows that are not homogeneous in the principle curvatures, finding various situations where closed, convex curvature-pinched hypersurfaces contracting self-similarly are necessarily spheres. Keywords Curvature flow · Parabolic partial differential equation · Self-similar solution Mathematics Subject Classification 53C44
1 Introduction In [21], the author considered contracting self-similar solutions of fully nonlinear curvature contraction flows whose speeds were homogeneous functions of the principle curvatures. This work extended earlier results of Huisken for the mean curvature flow [18], where it was shown that a compact hypersurface with nonnegative mean curvature contracting self-similarly under the mean curvature flow is necessarily a sphere. In the case of surfaces of dimension 2, the condition of nonnegative mean curvature was replaced by the requirements that the surface be embedded and have genus 0 by Brendle [8].
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James A. McCoy [email protected] Priority Research Centre Computer Assisted Research Mathematics and Applications, School of Mathematical and Physical Sciences, University of Newcastle, University Drive, Callaghan, NSW 2308, Australia
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J. A. McCoy
For a survey of more general self-similar shrinkers of the mean curvature flow we refer to [13]. For other flows, we refer the reader to the discussion in [21], noting that few results are available apart from those for flows by powers of the Gauss curvature. Let us here briefly update recent developments. In [24] the author, together with Mofarreh and V-M Wheeler extended the results of [2] for contracting surfaces to the case of axially symmetric hypersurfaces, including corresponding characterisation results for when axially symmetric hypersurfaces contracting self-similarly are necessarily spheres. In [9], the authors proved that closed, strictly convex hypersurfaces 1 of the Gauss curvature are spheres if contracting self-similarly by powers α ≥ n+2 the inequality is strict, or ellipses in the equality case (see also [12] for some powers). This removes the extra conditions of [21,24] required to conclude the hypersurfaces are spheres in this case. The result was also shown slightly earlier with an additional symmetry assumption in [5]. The result of [9] was generalised to powers α ≥ k1 of the elementary symmetric functions σk of the principle curvatures, 1 ≤ k ≤ n − 1 in [16], showing closed, strictly convex hypersurface
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