Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature

Authors:
Ben Andrews and James McCoy

Journal:
Trans. Amer. Math. Soc. **364** (2012), 3427-3447

MSC (2010):
Primary 53C44, 35K55; Secondary 53C40, 53C21, 58J35

DOI:
https://doi.org/10.1090/S0002-9947-2012-05375-X

Published electronically:
March 8, 2012

MathSciNet review:
2901219

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit at infinity. We prove that the ratio of the circumradius to the inradius is bounded by a function of the circumradius with limit at zero. We apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one. For smooth, strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point, we prove that solutions remain smooth and strictly convex and become spherical in shape while contracting to points in finite time.

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Additional Information

**Ben Andrews**

Affiliation:
Centre for Mathematics and its Applications, Australian National University, ACT 0200 Australia

Email:
Ben.Andrews@maths.anu.edu.au

**James McCoy**

Affiliation:
Institute of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia

Email:
jamesm@uow.edu.au

DOI:
https://doi.org/10.1090/S0002-9947-2012-05375-X

Received by editor(s):
October 1, 2009

Received by editor(s) in revised form:
April 28, 2010

Published electronically:
March 8, 2012

Additional Notes:
This research was partially supported by Discovery Grant DP0556211 of the Australian Research Council and by a University of Wollongong Research Council small grant entitled “Flow of convex hypersurfaces to spheres by high powers of curvature”

Article copyright:
© Copyright 2012
American Mathematical Society