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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: March 8, 2012
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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 $ 1$ at infinity. We prove that the ratio of the circumradius to the inradius is bounded by a function of the circumradius with limit $ 1$ 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: http://dx.doi.org/10.1090/S0002-9947-2012-05375-X
PII: S 0002-9947(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