Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature
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- by Ben Andrews and James McCoy PDF
- Trans. Amer. Math. Soc. 364 (2012), 3427-3447 Request permission
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.References
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Additional Information
- Ben Andrews
- Affiliation: Centre for Mathematics and its Applications, Australian National University, ACT 0200 Australia
- MR Author ID: 317229
- ORCID: 0000-0002-6507-0347
- Email: Ben.Andrews@maths.anu.edu.au
- James McCoy
- Affiliation: Institute of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia
- MR Author ID: 724395
- Email: jamesm@uow.edu.au
- 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”
- © Copyright 2012 American Mathematical Society
- 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
- MathSciNet review: 2901219