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Curvature contraction flows in the sphere


Author: James A. McCoy
Journal: Proc. Amer. Math. Soc. 146 (2018), 1243-1256
MSC (2010): Primary 35K55, 53C44
DOI: https://doi.org/10.1090/proc/13831
Published electronically: October 30, 2017
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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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Additional Information

James A. McCoy
Affiliation: Institute for Mathematics and its Applications, University of Wollongong, Northfields Avenue, Wollongong, NSW 2522, Australia
Email: jamesm@uow.edu.au

DOI: https://doi.org/10.1090/proc/13831
Keywords: Curvature flow, parabolic partial differential equation, hypersurface, axial symmetry, spherical geometry
Received by editor(s): December 18, 2016
Received by editor(s) in revised form: May 8, 2017
Published electronically: October 30, 2017
Additional Notes: This work was completed while the author was supported by DP150100375 of the Australian Research Council. The author would like to thank Professor Graham Williams for his interest in this work and Professor Ben Andrews, Doctor Glen Wheeler and Doctor Valentina Wheeler for useful discussions. The author would also like to thank the anonymous referee whose suggestions have led to improvements in the article.
Communicated by: Lei Ni
Article copyright: © Copyright 2017 American Mathematical Society

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