Curvature contraction flows in the sphere
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- by James A. McCoy PDF
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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.References
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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
- MR Author ID: 724395
- Email: jamesm@uow.edu.au
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1243-1256
- MSC (2010): Primary 35K55, 53C44
- DOI: https://doi.org/10.1090/proc/13831
- MathSciNet review: 3750236