Curvature contraction flows in the sphere

McCoy, James

doi:10.1090/proc/13831

Curvature contraction flows in the sphere
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by James A. McCoy PDF

Proc. Amer. Math. Soc. 146 (2018), 1243-1256 Request permission

Abstract:

We show that convex surfaces in an ambient three-sphere contract to round points in finite time under fully nonlinear, degree one homogeneous curvature flows, with no concavity condition on the speed. The result extends to convex axially symmetric hypersurfaces of $\mathbb {S}^{n+1}$. Using a different pinching function we also obtain the analogous results for contraction by Gauss curvature.

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