Analysis & PDE

• Anal. PDE
• Volume 7, Number 2 (2014), 407-433.

Convexity estimates for hypersurfaces moving by convex curvature functions

Abstract

We consider the evolution of compact hypersurfaces by fully nonlinear, parabolic curvature flows for which the normal speed is given by a smooth, convex, degree-one homogeneous function of the principal curvatures. We prove that solution hypersurfaces on which the speed is initially positive become weakly convex at a singularity of the flow. The result extends the convexity estimate of Huisken and Sinestrari [Acta Math. 183:1 (1999), 45–70] for the mean curvature flow to a large class of speeds, and leads to an analogous description of “type-II” singularities. We remark that many of the speeds considered are positive on larger cones than the positive mean half-space, so that the result in those cases also applies to non-mean-convex initial data.

Article information

Source
Anal. PDE, Volume 7, Number 2 (2014), 407-433.

Dates
Accepted: 23 July 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731494

Digital Object Identifier
doi:10.2140/apde.2014.7.407

Mathematical Reviews number (MathSciNet)
MR3218814

Zentralblatt MATH identifier
1294.53058

Citation

Andrews, Ben; Langford, Mathew; McCoy, James. Convexity estimates for hypersurfaces moving by convex curvature functions. Anal. PDE 7 (2014), no. 2, 407--433. doi:10.2140/apde.2014.7.407. https://projecteuclid.org/euclid.apde/1513731494

References

• R. Alessandroni and C. Sinestrari, “Convexity estimates for a nonhomogeneous mean curvature flow”, Math. Z. 266:1 (2010), 65–82.
• B. Andrews, “Contraction of convex hypersurfaces in Euclidean space”, Calc. Var. Partial Differential Equations 2:2 (1994), 151–171.
• B. Andrews, “Harnack inequalities for evolving hypersurfaces”, Math. Zeitschrift 217:2 (1994), 179–197.
• B. Andrews, “Fully nonlinear parabolic equations in two space variables”, preprint, 2004.
• B. Andrews, “Pinching estimates and motion of hypersurfaces by curvature functions”, J. Reine Angew. Math. 608 (2007), 17–33.
• B. Andrews, “Moving surfaces by non-concave curvature functions”, Calc. Var. Partial Differential Equations 39:3-4 (2010), 649–657.
• B. Andrews and C. Baker, “Mean curvature flow of pinched submanifolds to spheres”, J. Differential Geom. 85:3 (2010), 357–395.
• B. Andrews and C. Hopper, The Ricci flow in Riemannian geometry: a complete proof of the differentiable 1/4-pinching sphere theorem, Lecture Notes in Math. 2011, Springer, Heidelberg, 2011.
• B. Andrews, M. Langford, and J. A. McCoy, “Convexity estimates for surfaces moving by curvature functions”, preprint, 2012. To appear in J. Differential Geom.
• B. Andrews, M. Langford, and J. A. McCoy, “Non-collapsing in fully non-linear curvature flows”, Ann. Inst. H. Poincaré Anal. Non Linéaire 30:1 (2013), 23–32.
• B. Andrews, J. A. McCoy, and Y. Zheng, “Contracting convex hypersurfaces by curvature”, Calc. Var. Partial Differential Equations 47:3-4 (2013), 611–665.
• R. C. Baker, The mean curvature flow of submanifolds of high codimension, thesis, Australian National University, Canberra, 2010.
• R. C. Baker, “A partial classification of type I singularities of the mean curvature flow in high codimension”, preprint, 2011.
• B. Chow, “Deforming convex hypersurfaces by the $n$th root of the Gaussian curvature”, J. Differential Geom. 22:1 (1985), 117–138.
• B. Chow, “Deforming convex hypersurfaces by the square root of the scalar curvature”, Invent. Math. 87:1 (1987), 63–82.
• K. Ecker and G. Huisken, “Immersed hypersurfaces with constant Weingarten curvature”, Math. Ann. 283:2 (1989), 329–332.
• L. C. Evans, “Classical solutions of fully nonlinear, convex, second-order elliptic equations”, Comm. Pure Appl. Math. 35:3 (1982), 333–363.
• C. Gerhardt, “Flow of nonconvex hypersurfaces into spheres”, J. Differential Geom. 32:1 (1990), 299–314.
• C. Gerhardt, Curvature problems, Series in Geometry and Topology 39, International Press, Somerville, MA, 2006.
• Y. Giga and S. Goto, “Geometric evolution of phase-boundaries”, pp. 51–65 in On the evolution of phase boundaries (Minneapolis, MN, 1990–1991), edited by M. E. Gurtin and G. B. McFadden, IMA Vol. Math. Appl. 43, Springer, New York, 1992.
• G. Glaeser, “Fonctions composées différentiables”, Ann. of Math. $(2)$ 77 (1963), 193–209.
• R. S. Hamilton, “Three-manifolds with positive Ricci curvature”, J. Differential Geom. 17:2 (1982), 255–306.
• R. S. Hamilton, “Four-manifolds with positive curvature operator”, J. Differential Geom. 24:2 (1986), 153–179.
• R. S. Hamilton, “The formation of singularities in the Ricci flow”, pp. 7–136 in Surveys in differential geometry (Cambridge, MA, 1993), vol. II, edited by C. C. Hsiung and S.-T. Yau, International Press, Cambridge, MA, 1995.
• R. S. Hamilton, “Harnack estimate for the mean curvature flow”, J. Differential Geom. 41:1 (1995), 215–226.
• G. Huisken, “Flow by mean curvature of convex surfaces into spheres”, J. Differential Geom. 20:1 (1984), 237–266.
• G. Huisken, “Asymptotic behavior for singularities of the mean curvature flow”, J. Differential Geom. 31:1 (1990), 285–299.
• G. Huisken, “Local and global behaviour of hypersurfaces moving by mean curvature”, pp. 175–191 in Differential geometry, 1: Partial differential equations on manifolds (Los Angeles, CA, 1990), edited by R. Greene and S.-T. Yau, Proc. Sympos. Pure Math. 54, Amer. Math. Soc., Providence, RI, 1993.
• G. Huisken and C. Sinestrari, “Convexity estimates for mean curvature flow and singularities of mean convex surfaces”, Acta Math. 183:1 (1999), 45–70.
• G. Huisken and C. Sinestrari, “Mean curvature flow singularities for mean convex surfaces”, Calc. Var. Partial Differential Equations 8:1 (1999), 1–14.
• N. V. Krylov, “\cyr Ogranichenno neodnorodnye e1llipticheskie i parabolicheskie uravneniya”, Izv. Akad. Nauk SSSR Ser. Mat. 46:3 (1982), 487–523. Translated as in Math. USSR-Izv. 20:3 (1983), 459–492.
• J. A. McCoy, “Mixed volume preserving curvature flows”, Calc. Var. Partial Differential Equations 24:2 (2005), 131–154.
• J. A. McCoy, “Self-similar solutions of fully nonlinear curvature flows”, Ann. Sc. Norm. Super. Pisa Cl. Sci. $(5)$ 10:2 (2011), 317–333.
• J. McCoy, F. Mofarreh, and G. Williams, “Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions”, Ann. Mat. Pura Appl. (2014). To appear; posted online March 2013.
• J. H. Michael and L. M. Simon, “Sobolev and mean-value inequalities on generalized submanifolds of $\R^n$”, Comm. Pure Appl. Math. 26 (1973), 361–379.
• F. Schulze, “Convexity estimates for flows by powers of the mean curvature”, Ann. Sc. Norm. Super. Pisa Cl. Sci. $(5)$ 5:2 (2006), 261–277.
• G. Stampacchia, Èquations elliptiques du second ordre à coefficients discontinus, Séminaire de Mathématiques Supérieures 16, Les Presses de l'Université de Montréal, Montréal, QC, 1966.