Evolution of hypersurfaces by powers of the scalar curvature
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 3, p. 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
@article{ASNSP_2010_5_9_3_541_0,
author = {Alessandroni, Roberta and Sinestrari, Carlo},
title = {Evolution of hypersurfaces by powers of the scalar curvature},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {3},
year = {2010},
pages = {541-571},
zbl = {1248.53047},
mrnumber = {2722655},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_3_541_0}
}

Alessandroni, Roberta; Sinestrari, Carlo. Evolution of hypersurfaces by powers of the scalar curvature. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 3, pp. 541-571. http://www.numdam.org/item/ASNSP_2010_5_9_3_541_0/

[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 1385524 | Zbl 0805.35048

[3] B. Andrews, Gauss curvature flow: the fate of rolling stones, Invent. Math. 138 (1999), 151–161. | MR 1714339 | Zbl 0936.35080

[4] B. Andrews, Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 95 (2000), 1–36. | MR 1781612 | Zbl 1028.53072

[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 2729317 | Zbl 1203.53062

[7] B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math. 608 (2007), 17–33. | MR 2339467 | Zbl 1129.53044

[8] P. Bayard, Entire scalar curvature flow and hypersurfaces of constant scalar curvature in Minkowski space, Methods Appl. Anal. 16 (2009), 87–118. | MR 2556831 | Zbl 1180.53069

[9] B. Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Differential Geom. 22 (1985), 117–138. | MR 826427 | Zbl 0589.53005

[10] B. Chow, Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), 63–82. | MR 862712 | Zbl 0608.53005

[11] E. Di Benedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1–22. | MR 783531

[12] K. Ecker, “Regularity Theory for Mean Curvature Flow”, Birkhäuser, Boston, 2004. | MR 2024995 | Zbl 1058.53054

[13] C. Enz, The scalar curvature flow in lorentzian manifolds, Adv. Calc. Var. 1 (2008), 323–343. | MR 2458241 | Zbl 1162.53050

[14] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237–266. | MR 772132 | Zbl 0556.53001

[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 1731639 | Zbl 0942.35047

[16] G. Huisken and C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differential Equations 8 (1999), 1–14. | MR 1666878 | Zbl 0992.53052

[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 1719551 | Zbl 0992.53051

[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 1465184 | Zbl 0884.35001

[20] O. C. Schnürer, Surfaces contracting with speed ${\left|A\right|}^{2}$, J. Differential Geom. 71 (2005), 347–363. | MR 2198805 | Zbl 1101.53002

[21] F. Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Math. Z. 251 (2005), 721–733. | MR 2190140 | Zbl 1087.53062

[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. | Numdam | MR 2244700

[23] K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), 867–882. | MR 812353 | Zbl 0612.53005