Contracting convex hypersurfaces by curvature. (English) Zbl 1288.35292
Authors’ abstract: We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smooth, and the limiting shape at the final time is spherical. We provide a surprisingly large family of flows for which such results fail, by a variety of mechanisms: Uniformly convex hypersurfaces may become non-convex, and smooth ones may develop curvature singularities; even where this does not occur, non-uniformly convex regions and singular parts in the initial hypersurface may persist, including flat sides, ridges of infinite curvature, or ‘cylindrical’ regions where some of the principal curvatures vanish; such cylindrical regions may persist even if the speed is positive, and in such cases the hypersurface may even collapse to a line segment or higher-dimensional disc rather than to a point. We provide sufficient conditions for these various disasters to occur, and by avoiding these arrive at a class of flows for which arbitrary weakly convex initial hypersurfaces immediately become smooth and uniformly convex and contract to points.
Reviewer: Kungching Chang (Beijing)
MSC:
| 35K55 | Nonlinear parabolic equations |
| 35K45 | Initial value problems for second-order parabolic systems |
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
Keywords:
limiting shape; minimal barriers; collapse to a point; uniformly convex smooth initial hypersurfaces; curvature singularities; flat sides; collapse to a line segmentReferences:
| [1] | Andrews B.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Partial Differ. Equ. 2(2), 151-171 (1994) · Zbl 0805.35048 · doi:10.1007/BF01191340 |
| [2] | Andrews B.: Harnack inequalities for evolving hypersurfaces. Math. Z. 217(2), 179-197 (1994) · Zbl 0807.53044 · doi:10.1007/BF02571941 |
| [3] | Andrews B.: Monotone quantities and unique limits for evolving convex hypersurfaces. Int. Math. Res. Not. 20, 1001-1031 (1997) · Zbl 0892.53002 · doi:10.1155/S1073792897000640 |
| [4] | Andrews B.: Evolving convex curves. Calc. Var. Partial Differ. Equ. 7(4), 315-371 (1998) · Zbl 0931.53030 · doi:10.1007/s005260050111 |
| [5] | Andrews B.: Gauss curvature flow: the fate of the rolling stones. Invent. Math. 138(1), 151-161 (1999) · Zbl 0936.35080 · doi:10.1007/s002220050344 |
| [6] | Andrews B.: Motion of hypersurfaces by Gauss curvature. Pac. J. Math. 195(1), 1-34 (2000) · Zbl 1028.53072 · doi:10.2140/pjm.2000.195.1 |
| [7] | Andrews B.: Volume-preserving anisotropic mean curvature flow. Indiana Univ. Math. J. 50(2), 783-827 (2001) · Zbl 1047.53037 · doi:10.1512/iumj.2001.50.1853 |
| [8] | Andrews B.: Pinching estimates and motion of hypersurfaces by curvature functions. J. Reine Angew. Math. 608, 17-33 (2007) · Zbl 1129.53044 |
| [9] | Andrews B.: Moving surfaces by non-concave curvature functions. Calc. Var. Partial Differ. Equ. 39(3-4), 649-657 (2010) · Zbl 1203.53062 · doi:10.1007/s00526-010-0329-z |
| [10] | Andrews, B.: Fully nonlinear parabolic equations in two space variables. eprint, arXiv:math.DG/ 0402235v1 [math.AP] · Zbl 0882.35028 |
| [11] | Andrews B., McCoy J.: Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature. Trans. Am. Math. Soc. 364, 3427-3447 (2012) · Zbl 1277.53061 · doi:10.1090/S0002-9947-2012-05375-X |
| [12] | Bellettini G., Novaga M.: Minimal barriers for geometric evolutions. J. Differ. Equ. 139(1), 76-103 (1997) · Zbl 0882.35028 · doi:10.1006/jdeq.1997.3288 |
| [13] | Bellettini G., Novaga M.: Comparison results between minimal barriers and viscosity solutions for geometric evolutions. Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 26, 97-131 (1998) · Zbl 0904.35041 |
| [14] | Bian B., Guan P.: A microscopic convexity principle for nonlinear partial differential equations. Invent. Math. 177(2), 307-335 (2009) · Zbl 1168.35010 · doi:10.1007/s00222-009-0179-5 |
| [15] | Bony, J.-M.: Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann Inst. Fourier (Grenoble) 19(fasc. 1), 277-304 (1969) (xii (French, with English summary)) · Zbl 0176.09703 |
| [16] | Caffarelli L., Nirenberg L., Spruck J.: The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math. 155(3-4), 261-301 (1985) · Zbl 0654.35031 · doi:10.1007/BF02392544 |
| [17] | Caffarelli L., Guan P., Ma X.-N.: A constant rank theorem for solutions of fully nonlinear elliptic equations. Commun. Pure Appl. Math. 60(12), 1769-1791 (2007) · Zbl 1191.35127 · doi:10.1002/cpa.20197 |
| [18] | Caputo M.C., Daskalopoulos P.: Highly degenerate harmonic mean curvature flow. Calc. Var. Partial Differ. Equ. 35(3), 365-384 (2009) · Zbl 1179.35174 · doi:10.1007/s00526-008-0209-y |
| [19] | Caputo M.C., Daskalopoulos P., Sesum N.: On the evolution of convex hypersurfaces by the Qk flow. Commun. Partial Differ. Equ. 35(3), 415-442 (2010) · Zbl 1187.53068 · doi:10.1080/03605300903296314 |
| [20] | Chen Y.G., Giga Y., Goto S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33(3), 749-786 (1991) · Zbl 0696.35087 |
| [21] | Chow B.: Deforming convex hypersurfaces by the nth root of the Gaussian curvature. J. Differ. Geom. 22(1), 117-138 (1985) · Zbl 0589.53005 |
| [22] | Chow B.: Deforming convex hypersurfaces by the square root of the scalar curvature. Invent. Math. 87(1), 63-82 (1987) · Zbl 0608.53005 · doi:10.1007/BF01389153 |
| [23] | Daskalopoulos P., Hamilton R.: The free boundary in the Gauss curvature flow with flat sides. J. Reine Angew. Math. 510, 187-227 (1999) · Zbl 0931.53031 |
| [24] | Dieter S.: Nonlinear degenerate curvature flows for weakly convex hypersurfaces. Calc. Var. Partial Differ. Equ. 22(2), 229-251 (2005) · Zbl 1076.53079 · doi:10.1007/s00526-004-0279-4 |
| [25] | Ecker K., Huisken G.: Immersed hypersurfaces with constant Weingarten curvature. Math. Ann. 283(2), 329-332 (1989) · Zbl 0643.53043 · doi:10.1007/BF01446438 |
| [26] | Evans L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35(3), 333-363 (1982) · Zbl 0469.35022 · doi:10.1002/cpa.3160350303 |
| [27] | Evans L.C., Spruck J.: Motion of level sets by mean curvature. I. J. Differ. Geom. 33((3), 635-681 (1991) · Zbl 0726.53029 |
| [28] | Gerhardt C.: Closed Weingarten hypersurfaces in Riemannian manifolds. J. Differ. Geom. 43(3), 612-641 (1996) · Zbl 0861.53058 |
| [29] | Giga Y., Goto S., Ishii H., Sato M.-H.: Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40(2), 443-470 (1991) · Zbl 0836.35009 · doi:10.1512/iumj.1991.40.40023 |
| [30] | Glaeser G.: Fonctions composées différentiables, French. Ann. Math. 77(2), 193-209 (1963) · Zbl 0106.31302 · doi:10.2307/1970204 |
| [31] | Goto S.: Generalized motion of hypersurfaces whose growth speed depends superlinearly on the curvature tensor. Differ. Integral Equ. 7(2), 323-343 (1994) · Zbl 0808.35007 |
| [32] | Hamilton R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom 17(2), 255-306 (1982) · Zbl 0504.53034 |
| [33] | Hamilton, R.S.: Worn stones with flat sides. A tribute to Ilya Bakelman (College Station, TX, 1993). Discourses in Mathematics and Its Applications, vol. 3, pp. 69-78. Texas A & M Univ., College Station (1994) · Zbl 1203.53062 |
| [34] | Han Q.: Deforming convex hypersurfaces by curvature functions. Analysis 17(2-3), 113-127 (1997) · Zbl 0992.53009 |
| [35] | Hill, C.D.: A sharp maximum principle for degenerate elliptic-parabolic equations. Indiana Univ. Math. J. 20, 213-229 (1970/1971) · Zbl 0205.10402 |
| [36] | Huisken G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237-266 (1984) · Zbl 0556.53001 |
| [37] | Ilmanen, T.: The level-set flow on a manifold, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990). In: Proceedings of Symposia in Pure Mathematics, vol. 54, pp. 193-204. American Mathematical Society, Providence (1993) · Zbl 0827.53014 |
| [38] | Ishii H., Souganidis P.: Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor. Tohoku Math. J. 47(2), 227-250 (1995) · Zbl 0837.35066 · doi:10.2748/tmj/1178225593 |
| [39] | Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations. Izv. Akad. Nauk SSSR Ser. Mat. 46(3), 487-523, 670 (1982) (Russian) · Zbl 0511.35002 |
| [40] | Lieberman G.M.: Second Order Parabolic Differential Equations. World Scientific Publishing Co. Inc., River Edge, NJ (1996) · Zbl 0884.35001 · doi:10.1142/3302 |
| [41] | McCoy J.A.: The mixed volume preserving mean curvature flow. Math. Z. 246(1-2), 155-166 (2004) · Zbl 1062.53057 · doi:10.1007/s00209-003-0592-1 |
| [42] | McCoy J.A.: Mixed volume preserving curvature flows. Calc. Var. Partial Differ. Equ. 24(2), 131-154 (2005) · Zbl 1079.53099 · doi:10.1007/s00526-004-0316-3 |
| [43] | Schwarz G.W.: Smooth functions invariant under the action of a compact Lie group. Topology 14, 63-68 (1975) · Zbl 0297.57015 · doi:10.1016/0040-9383(75)90036-1 |
| [44] | Smoczyk K.: Starshaped hypersurfaces and the mean curvature flow. Manuscr. Math. 95(2), 225-236 (1998) · Zbl 0903.53039 · doi:10.1007/s002290050025 |
| [45] | Tso K.: Deforming a hypersurface by its Gauss-Kronecker curvature. Commun. Pure Appl. Math. 38(6), 867-882 (1985) · Zbl 0612.53005 · doi:10.1002/cpa.3160380615 |
| [46] | Urbas J.I.E.: An expansion of convex hypersurfaces. J. Differ. Geom. 33(1), 91-125 (1991) · Zbl 0746.53006 |
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