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 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 . Finally we show that self-similar hypersurfaces with sufficiently pinched principal curvatures, depending on the flow speed, are again necessarily spheres.
@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}, pages = {317--333}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 10}, number = {2}, year = {2011}, mrnumber = {2856150}, zbl = {1234.53018}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2011_5_10_2_317_0/} }
TY - JOUR AU - McCoy, James Alexander TI - Self-similar solutions of fully nonlinear curvature flows JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2011 SP - 317 EP - 333 VL - 10 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2011_5_10_2_317_0/ LA - en ID - ASNSP_2011_5_10_2_317_0 ER -
%0 Journal Article %A McCoy, James Alexander %T Self-similar solutions of fully nonlinear curvature flows %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2011 %P 317-333 %V 10 %N 2 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2011_5_10_2_317_0/ %G en %F 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, Série 5, Tome 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 | Zbl
[2] A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. V, Vestnik Leningrad. Univ. 13 (1958), 5–8. | MR | Zbl
[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 | Zbl
[4] B. H. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994), 151–171. | MR | Zbl
[5] B. H. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geom. 43 (1996), 207–230. | MR | Zbl
[6] B. H. Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), 151–161. | MR | Zbl
[7] B. H. Andrews, Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 195 (2000), 1–34. | MR | Zbl
[8] B. H. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math. 608 (2007), 17–33. | MR | Zbl
[9] B. H. Andrews, Moving surfaces by non-concave curvature functions, Calc. Var. Partial Differential Equations 39 (2010), 649–657. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[14] E. Calabi, Complete affine hyperspheres, I, Sympos. Math. 10 (1971/72), 19–38 (1972). | MR | Zbl
[15] B. Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Differential Geom. 22 (1985), 117–138. | MR | Zbl
[16] B. Chow, Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), 63–82. | EuDML | MR | Zbl
[17] C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom. 32 (1990), 299–314. | MR | Zbl
[18] C. Gerhardt, “Curvature Problems”, Series in Geometry and Topology, Vol. 39, International Press, Somerville, 2006. | MR | Zbl
[19] Q. Han, Deforming convex hypersurfaces by curvature functions, Analysis 17 (1997), 113–127. | MR | Zbl
[20] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237–266. | MR | Zbl
[21] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), 285–299. | MR | Zbl
[22] G. M. Lieberman, “Second Order Parabolic Differential Equations”, World Scientific, Singapore, 1996. | MR | Zbl
[23] O. C. Schnürer, Surfaces contracting with speed , J. Differential Geom. 71 (2005), 347–363. | MR | Zbl
[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. | EuDML | Numdam | MR | Zbl
[25] K. Smoczyk, Harnack inequalities for curvature flows depending on mean curvature, New York J. Math. 3 (1997), 103–118 (electronic). | EuDML | MR | Zbl
[26] K. Smoczyk, Self-shrinkers of the mean curvature flow in arbitrary codimension, Int. Math. Res. Not. 48 (2005), 2983–3004. | MR | Zbl
[27] K.-S. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), 867–882. | MR | Zbl
[28] J. I. E. Urbas, An expansion of convex hypersurfaces, J. Differential Geom. 33 (1991), 91–125. | MR | Zbl
[29] J. I. E. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z. 205 (1990), 355–372. | EuDML | MR | Zbl