Evolution of hypersurfaces by powers of the scalar curvature

Alessandroni, Roberta; Sinestrari, Carlo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 3, pp. 541-571.

We study the evolution of a closed hypersurface of the euclidean space by a flow whose speed is given by a power of the scalar curvature. We prove that, if the initial shape is convex and satisfies a suitable pinching condition, the solution shrinks to a point in finite time and converges to a sphere after rescaling. We also give an example of a nonconvex hypersurface which develops a neckpinch singularity.

Classification : 53C44, 35K55, 58J35, 35B40

[1] R. Alessandroni, “Evolution of Hypersurfaces by Curvature Functions”, PhD Thesis, Università di Roma “Tor Vergata”, 2008.

[2] B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994), 151–171. | MR | Zbl

[3] B. Andrews, Gauss curvature flow: the fate of rolling stones, Invent. Math. 138 (1999), 151–161. | MR | Zbl

[4] B. Andrews, Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 95 (2000), 1–36. | MR | Zbl

[5] B. Andrews, Fully nonlinear parabolic equations in two space variables, arXiv:math.AP/0402235 vl (2004).

[6] B. Andrews, Moving surfaces by non-concave curvature functions, Calc. Var. Partial Differential Equations, to appear. | MR | Zbl

[7] B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math. 608 (2007), 17–33. | MR | Zbl

[8] P. Bayard, Entire scalar curvature flow and hypersurfaces of constant scalar curvature in Minkowski space, Methods Appl. Anal. 16 (2009), 87–118. | MR | Zbl

[9] B. Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Differential Geom. 22 (1985), 117–138. | MR | Zbl

[10] B. Chow, Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), 63–82. | EuDML | MR | Zbl

[11] E. Di Benedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1–22. | EuDML | MR

[12] K. Ecker, “Regularity Theory for Mean Curvature Flow”, Birkhäuser, Boston, 2004. | MR | Zbl

[13] C. Enz, The scalar curvature flow in lorentzian manifolds, Adv. Calc. Var. 1 (2008), 323–343. | MR | Zbl

[14] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237–266. | MR | Zbl

[15] G. Huisken and A. Polden, Geometric evolution equation for hypersurfaces. In: “Calculus of Variations and Geometric Evolution Problems (CIME, Cetraro, 1996)”, S. Hildebrandt et al. (eds.), Lect. Notes Math., Vol. 1713, Springer, Berlin, 1996, 45–84. | MR | Zbl

[16] G. Huisken and C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differential Equations 8 (1999), 1–14. | MR | Zbl

[17] G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999), 45–70. | MR | Zbl

[18] N. V. Krylov, “Nonlinear Elliptic and Parabolic Equations of Second Order”, D. Reidel, Dordrecht, 1987.

[19] G. M. Lieberman, “Second Order Parabolic Differential Equations”, World Scientific, River Edge, NJ, 1996. | MR | Zbl

[20] O. C. Schnürer, Surfaces contracting with speed ${\left|A\right|}^2$, J. Differential Geom. 71 (2005), 347–363. | MR | Zbl

[21] F. Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Math. Z. 251 (2005), 721–733. | MR | Zbl

[22] F. Schulze, Convexity estimates for flows by powers of the mean curvature, with an appendix by O. C. Schnürer and F. Schulze, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 5 (2006), 261–277. | EuDML | Numdam | MR

[23] K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), 867–882. | MR | Zbl