Self-similar solutions of fully nonlinear curvature flows
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, p. 317-333

We consider closed hypersurfaces which shrink self-similarly under a natural class of fully nonlinear curvature flows. For those flows in our class with speeds homogeneous of degree $1$ and either convex or concave, we show that the only such hypersurfaces are shrinking spheres. In the setting of convex hypersurfaces, we show under a weaker second derivative condition on the speed that again only shrinking spheres are possible. For surfaces this result is extended in some cases by a different method to speeds of homogeneity greater than $1$. Finally we show that self-similar hypersurfaces with sufficiently pinched principal curvatures, depending on the flow speed, are again necessarily spheres.

Published online : 2018-08-07
Classification:  53C44,  35J60
@article{ASNSP_2011_5_10_2_317_0,
author = {McCoy, James Alexander},
title = {Self-similar solutions of fully nonlinear curvature flows},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 10},
number = {2},
year = {2011},
pages = {317-333},
zbl = {1234.53018},
mrnumber = {2856150},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2011_5_10_2_317_0}
}

McCoy, James Alexander. Self-similar solutions of fully nonlinear curvature flows. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, pp. 317-333. http://www.numdam.org/item/ASNSP_2011_5_10_2_317_0/

[1] U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Differential Geom. 23 (1986), 175–196. | MR 845704 | Zbl 0592.53002

[2] A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. V, Vestnik Leningrad. Univ. 13 (1958), 5–8. | MR 102114 | Zbl 0119.16603

[3] K. Anada, Contraction of surfaces by harmonic mean curvature flows and nonuniqueness of their self similar solutions, Calc. Var. Partial Differential Equations 12 (2001), 109–116. | MR 1821233 | Zbl 1005.53004

[4] B. H. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994), 151–171. | MR 1385524 | Zbl 0805.35048

[5] B. H. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geom. 43 (1996), 207–230. | MR 1424425 | Zbl 0858.53005

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

[7] B. H. Andrews, Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 195 (2000), 1–34. | MR 1781612 | Zbl 1028.53072

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

[9] B. H. Andrews, Moving surfaces by non-concave curvature functions, Calc. Var. Partial Differential Equations 39 (2010), 649–657. | MR 2729317 | Zbl 1203.53062

[10] B. H. Andrews and J. A. McCoy, Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature, Trans. Amer. Math. Soc., to appear. | MR 2901219 | Zbl 1277.53061

[11] B. H. Andrews, J. A. McCoy and Z. Yu, Contraction of nonsmooth convex hypersurfaces into spheres, available at arxiv.org/abs/1104.0756.

[12] S. Angenent, Shrinking doughnuts, Progr. Nonlinear Differential Equations Appl. 7 (1992), Birkhäuser, 21–38. | MR 1167827 | Zbl 0762.53028

[13] E. Cabezas-Rivas and C. Sinestrari, Volume-preserving flow by powers of the mth mean curvature, Calc. Var. Partial Differential Equations 38 (2010), 441–469. | MR 2647128 | Zbl 1197.53082

[14] E. Calabi, Complete affine hyperspheres, I, Sympos. Math. 10 (1971/72), 19–38 (1972). | MR 365607 | Zbl 0252.53008

[15] 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

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

[17] C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom. 32 (1990), 299–314. | MR 1064876 | Zbl 0708.53045

[18] C. Gerhardt, “Curvature Problems”, Series in Geometry and Topology, Vol. 39, International Press, Somerville, 2006. | MR 2284727 | Zbl 1131.53001

[19] Q. Han, Deforming convex hypersurfaces by curvature functions, Analysis 17 (1997), 113–127. | MR 1486359 | Zbl 0992.53009

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

[21] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), 285–299. | MR 1030675 | Zbl 0694.53005

[22] G. M. Lieberman, “Second Order Parabolic Differential Equations”, World Scientific, Singapore, 1996. | MR 1465184 | Zbl 0884.35001

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

[24] F. Schulze, Convexity estimates for flows by powers of the mean curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), 261–277. | Numdam | MR 2244700 | Zbl 1150.53024

[25] K. Smoczyk, Harnack inequalities for curvature flows depending on mean curvature, New York J. Math. 3 (1997), 103–118 (electronic). | MR 1480081 | Zbl 0897.53032

[26] K. Smoczyk, Self-shrinkers of the mean curvature flow in arbitrary codimension, Int. Math. Res. Not. 48 (2005), 2983–3004. | MR 2189784 | Zbl 1085.53059

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

[28] J. I. E. Urbas, An expansion of convex hypersurfaces, J. Differential Geom. 33 (1991), 91–125. | MR 1085136 | Zbl 0746.53006

[29] J. I. E. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z. 205 (1990), 355–372. | MR 1082861 | Zbl 0691.35048