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

MathSciNet review:
2901219

Full-text PDF

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.

**[A]**Roberta Alessandroni,*Evolution of hypersurfaces by curvature functions*, Ph.D. Thesis, Università degli studi di Roma ``Tor Vergata'', 2008,`http://dspace.uniroma2.it/dspace/handle/2108/661`.**[AS]**Roberta Alessandroni and Carlo Sinestrari,*Evolution of hypersurfaces by powers of the scalar curvature*, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)**9**(2010), no. 3, 541–571. MR**2722655****[A1]**Ben Andrews,*Contraction of convex hypersurfaces in Euclidean space*, Calc. Var. Partial Differential Equations**2**(1994), no. 2, 151–171. MR**1385524**, 10.1007/BF01191340**[A2]**Ben Andrews,*Gauss curvature flow: the fate of the rolling stones*, Invent. Math.**138**(1999), no. 1, 151–161. MR**1714339**, 10.1007/s002220050344**[A3]**Ben Andrews,*Pinching estimates and motion of hypersurfaces by curvature functions*, J. Reine Angew. Math.**608**(2007), 17–33. MR**2339467**, 10.1515/CRELLE.2007.051**[A4]**Ben Andrews,*Moving surfaces by non-concave curvature functions*, Calc. Var. Partial Differential Equations**39**(2010), no. 3-4, 649–657. MR**2729317**, 10.1007/s00526-010-0329-z**[CRS]**Esther Cabezas-Rivas and Carlo Sinestrari,*Volume-preserving flow by powers of the 𝑚th mean curvature*, Calc. Var. Partial Differential Equations**38**(2010), no. 3-4, 441–469. MR**2647128**, 10.1007/s00526-009-0294-6**[Ca]**Luis A. Caffarelli,*Interior a priori estimates for solutions of fully nonlinear equations*, Ann. of Math. (2)**130**(1989), no. 1, 189–213. MR**1005611**, 10.2307/1971480**[Ch1]**Bennett Chow,*Deforming convex hypersurfaces by the 𝑛th root of the Gaussian curvature*, J. Differential Geom.**22**(1985), no. 1, 117–138. MR**826427****[Ch2]**Bennett Chow,*Deforming convex hypersurfaces by the square root of the scalar curvature*, Invent. Math.**87**(1987), no. 1, 63–82. MR**862712**, 10.1007/BF01389153**[Co]**Heinz Otto Cordes,*Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen*, Math. Ann.**131**(1956), 278–312 (German). MR**0091400****[E]**Lawrence C. Evans,*Classical solutions of fully nonlinear, convex, second-order elliptic equations*, Comm. Pure Appl. Math.**35**(1982), no. 3, 333–363. MR**649348**, 10.1002/cpa.3160350303**[G]**Georges Glaeser,*Fonctions composées différentiables*, Ann. of Math. (2)**77**(1963), 193–209 (French). MR**0143058****[H1]**Gerhard Huisken,*Flow by mean curvature of convex surfaces into spheres*, J. Differential Geom.**20**(1984), no. 1, 237–266. MR**772132****[H2]**Gerhard Huisken,*The volume preserving mean curvature flow*, J. Reine Angew. Math.**382**(1987), 35–48. MR**921165**, 10.1515/crll.1987.382.35**[K]**N. V. Krylov,*Boundedly inhomogeneous elliptic and parabolic equations*, Izv. Akad. Nauk SSSR Ser. Mat.**46**(1982), no. 3, 487–523, 670 (Russian). MR**661144****[KS]**N. V. Krylov and M. V. Safonov,*A property of the solutions of parabolic equations with measurable coefficients*, Izv. Akad. Nauk SSSR Ser. Mat.**44**(1980), no. 1, 161–175, 239 (Russian). MR**563790****[L]**Gary M. Lieberman,*Second order parabolic differential equations*, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR**1465184****[N]**L. Nirenberg,*On a generalization of quasi-conformal mappings and its application to elliptic partial differential equations*, Contributions to the theory of partial differential equations, Annals of Mathematics Studies, no. 33, Princeton University Press, Princeton, N. J., 1954, pp. 95–100. MR**0066532****[S1]**Rolf Schneider,*Convex bodies: the Brunn-Minkowski theory*, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR**1216521****[S2]**Oliver C. Schnürer,*Surfaces contracting with speed \vert𝐴\vert²*, J. Differential Geom.**71**(2005), no. 3, 347–363. MR**2198805****[S3]**Felix Schulze,*Convexity estimates for flows by powers of the mean curvature*, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)**5**(2006), no. 2, 261–277. MR**2244700****[S4]**Knut Smoczyk,*Starshaped hypersurfaces and the mean curvature flow*, Manuscripta Math.**95**(1998), no. 2, 225–236. MR**1603325**, 10.1007/s002290050025**[T]**Kaising Tso,*Deforming a hypersurface by its Gauss-Kronecker curvature*, Comm. Pure Appl. Math.**38**(1985), no. 6, 867–882. MR**812353**, 10.1002/cpa.3160380615

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
53C44,
35K55,
53C40,
53C21,
58J35

Retrieve articles in all journals with MSC (2010): 53C44, 35K55, 53C40, 53C21, 58J35

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