Selfshrinkers with a rotational symmetry
Authors:
Stephen Kleene and Niels Martin Møller
Journal:
Trans. Amer. Math. Soc. 366 (2014), 39433963
MSC (2010):
Primary 53C44
Published electronically:
March 26, 2014
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Abstract: In this paper we present a new family of noncompact properly embedded, selfshrinking, asymptotically conical, positive mean curvature ends that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and halfcylinder in , and any rotationally symmetric selfshrinking noncompact end belongs to our family. The proofs involve the global analysis of a cubicderivative quasilinear ODE. We also prove the following classification result: a given complete, embedded, selfshrinking hypersurface of revolution is either a hyperplane , the round cylinder of radius , the round sphere of radius , or is diffeomorphic to an (i.e. a ``doughnut'' as in the paper by Sigurd B. Angenent, 1992, which when is a torus). In particular, for selfshrinkers there is no direct analogue of the Delaunay unduloid family. The proof of the classification uses translation and rotation of pieces, replacing the method of moving planes in the absence of isometries.
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Ilmanen, and D.
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(2007c:53008), http://dx.doi.org/10.1073/pnas.0510379103
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H. Colding and William
P. Minicozzi II, The space of embedded minimal surfaces of fixed
genus in a 3manifold. I. Estimates off the axis for disks, Ann. of
Math. (2) 160 (2004), no. 1, 27–68. MR 2119717
(2006a:53004), http://dx.doi.org/10.4007/annals.2004.160.27
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Tobias
H. Colding and William
P. Minicozzi II, The space of embedded minimal surfaces of fixed
genus in a 3manifold. II. Multivalued graphs in disks, Ann. of Math.
(2) 160 (2004), no. 1, 69–92. MR 2119718
(2006a:53005), http://dx.doi.org/10.4007/annals.2004.160.69
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Tobias
H. Colding and William
P. Minicozzi II, The space of embedded minimal surfaces of fixed
genus in a 3manifold. III. Planar domains, Ann. of Math. (2)
160 (2004), no. 2, 523–572. MR 2123932
(2006e:53012), http://dx.doi.org/10.4007/annals.2004.160.523
 [CM5]
Tobias
H. Colding and William
P. Minicozzi II, The space of embedded minimal surfaces of fixed
genus in a 3manifold. IV. Locally simply connected, Ann. of Math. (2)
160 (2004), no. 2, 573–615. MR 2123933
(2006e:53013), http://dx.doi.org/10.4007/annals.2004.160.573
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T. H. Colding, W. P. Minicozzi II, The space of embedded minimal surfaces in a 3manifold. V. Fixed genus, preprint.
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T.H. Colding, W. P. Minicozzi II, Generic mean curvature flow I; generic singularities, preprint, http://arxiv.org/pdf/0908.3788.
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T.H. Colding, W.P. Minicozzi II, Generic mean curvature flow II; dynamics of a closed smooth singularity, in preparation.
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C.
De Coster and P.
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(2001j:34023)
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Klaus
Ecker, Regularity theory for mean curvature flow, Progress in
Nonlinear Differential Equations and their Applications, 57,
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(2005b:53108)
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MiHo
Giga, Yoshikazu
Giga, and Jürgen
Saal, Nonlinear partial differential equations, Progress in
Nonlinear Differential Equations and their Applications, 79,
Birkhäuser Boston, Inc., Boston, MA, 2010. Asymptotic behavior of
solutions and selfsimilar solutions. MR 2656972
(2011f:35003)
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Wuyi
Hsiang and H.
Blaine Lawson Jr., Minimal submanifolds of low cohomogeneity,
J. Differential Geometry 5 (1971), 1–38. MR 0298593
(45 #7645)
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Gerhard
Huisken, Asymptotic behavior for singularities of the mean
curvature flow, J. Differential Geom. 31 (1990),
no. 1, 285–299. MR 1030675
(90m:53016)
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Gerhard
Huisken, Flow by mean curvature of convex surfaces into
spheres, J. Differential Geom. 20 (1984), no. 1,
237–266. MR
772132 (86j:53097)
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Gerhard
Huisken, Local and global behaviour of hypersurfaces moving by mean
curvature, Differential geometry: partial differential equations on
manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54,
Amer. Math. Soc., Providence, RI, 1993, pp. 175–191. MR 1216584
(94c:58037)
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T. Ilmanen, unpublished notes. Referenced as [21] in [4].
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Nikolaos
Kapouleas, Complete embedded minimal surfaces of finite total
curvature, J. Differential Geom. 47 (1997),
no. 1, 95–169. MR 1601434
(99a:53008)
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Katsuei
Kenmotsu, Surfaces with constant mean curvature, Translations
of Mathematical Monographs, vol. 221, American Mathematical Society,
Providence, RI, 2003. Translated from the 2000 Japanese original by
Katsuhiro Moriya and revised by the author. MR 2013507
(2004m:53014)
 [Ng]
Xuan
Hien Nguyen, Translating tridents, Comm. Partial Differential
Equations 34 (2009), no. 13, 257–280. MR 2512861
(2010c:53100), http://dx.doi.org/10.1080/03605300902768685
 [SS93]
H.
M. Soner and P.
E. Souganidis, Singularities and uniqueness of cylindrically
symmetric surfaces moving by mean curvature, Comm. Partial
Differential Equations 18 (1993), no. 56,
859–894. MR 1218522
(94c:53011), http://dx.doi.org/10.1080/03605309308820954
 [AAG95]
 Steven Altschuler, Sigurd B. Angenent, and Yoshikazu Giga, Mean curvature flow through singularities for surfaces of rotation, J. Geom. Anal. 5 (1995), no. 3, 293358. MR 1360824 (97j:58029), http://dx.doi.org/10.1007/BF02921800
 [Anc]
 Henri Anciaux, Two non existence results for the selfsimilar equation in Euclidean 3space, J. Geom. 96 (2009), no. 12, 110. MR 2659595 (2011f:53123), http://dx.doi.org/10.1007/s0002201000315
 [Ang]
 Sigurd B. Angenent, Shrinking doughnuts, (Gregynog, 1989) Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 2138. MR 1167827 (93d:58032)
 [AIC]
 S. Angenent, T. Ilmanen, and D. L. Chopp, A computed example of nonuniqueness of mean curvature flow in , Comm. Partial Differential Equations 20 (1995), no. 1112, 19371958. MR 1361726 (96k:58045), http://dx.doi.org/10.1080/03605309508821158
 [Ch]
 David L. Chopp, Computation of selfsimilar solutions for mean curvature flow, Experiment. Math. 3 (1994), no. 1, 115. MR 1302814 (95j:53006)
 [CM1]
 Tobias H. Colding and William P. Minicozzi II, Shapes of embedded minimal surfaces, Proc. Natl. Acad. Sci. USA 103 (2006), no. 30, 1110611111 (electronic). MR 2242650 (2007c:53008), http://dx.doi.org/10.1073/pnas.0510379103
 [CM2]
 Tobias H. Colding and William P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3manifold. I. Estimates off the axis for disks, Ann. of Math. (2) 160 (2004), no. 1, 2768. MR 2119717 (2006a:53004), http://dx.doi.org/10.4007/annals.2004.160.27
 [CM3]
 Tobias H. Colding and William P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3manifold. II. Multivalued graphs in disks, Ann. of Math. (2) 160 (2004), no. 1, 6992. MR 2119718 (2006a:53005), http://dx.doi.org/10.4007/annals.2004.160.69
 [CM4]
 Tobias H. Colding and William P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3manifold. III. Planar domains, Ann. of Math. (2) 160 (2004), no. 2, 523572. MR 2123932 (2006e:53012), http://dx.doi.org/10.4007/annals.2004.160.523
 [CM5]
 Tobias H. Colding and William P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3manifold. IV. Locally simply connected, Ann. of Math. (2) 160 (2004), no. 2, 573615. MR 2123933 (2006e:53013), http://dx.doi.org/10.4007/annals.2004.160.573
 [CM6]
 T. H. Colding, W. P. Minicozzi II, The space of embedded minimal surfaces in a 3manifold. V. Fixed genus, preprint.
 [CM7]
 T.H. Colding, W. P. Minicozzi II, Generic mean curvature flow I; generic singularities, preprint, http://arxiv.org/pdf/0908.3788.
 [CM8]
 T.H. Colding, W.P. Minicozzi II, Generic mean curvature flow II; dynamics of a closed smooth singularity, in preparation.
 [CH]
 C. De Coster and P. Habets, An overview of the method of lower and upper solutions for ODEs, (Lisbon, 1998) Progr. Nonlinear Differential Equations Appl., vol. 43, Birkhäuser Boston, Boston, MA, 2001, pp. 322. MR 1800611 (2001j:34023)
 [De]
 C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constante, J. Math. Pures et Appl. Ser. 1 (6) (1841), 309320.
 [Ec]
 Klaus Ecker, Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equations and their Applications, 57, Birkhäuser Boston Inc., Boston, MA, 2004. MR 2024995 (2005b:53108)
 [GGS10]
 MiHo Giga, Yoshikazu Giga, and Jürgen Saal, Nonlinear partial differential equations, Progress in Nonlinear Differential Equations and their Applications, 79, Birkhäuser Boston Inc., Boston, MA, 2010. Asymptotic behavior of solutions and selfsimilar solutions. MR 2656972 (2011f:35003)
 [HL]
 Wuyi Hsiang and H. Blaine Lawson Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geometry 5 (1971), 138. MR 0298593 (45 #7645)
 [Hu1]
 Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285299. MR 1030675 (90m:53016)
 [Hu2]
 Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237266. MR 772132 (86j:53097)
 [Hu3]
 Gerhard Huisken, Local and global behaviour of hypersurfaces moving by mean curvature, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 175191. MR 1216584 (94c:58037)
 [Il]
 T. Ilmanen, unpublished notes. Referenced as [21] in [4].
 [Ka]
 Nikolaos Kapouleas, Complete embedded minimal surfaces of finite total curvature, J. Differential Geom. 47 (1997), no. 1, 95169. MR 1601434 (99a:53008)
 [KK]
 Katsuei Kenmotsu, Surfaces with constant mean curvature, Translations of Mathematical Monographs, vol. 221, American Mathematical Society, Providence, RI, 2003. Translated from the 2000 Japanese original by Katsuhiro Moriya and revised by the author. MR 2013507 (2004m:53014)
 [Ng]
 Xuan Hien Nguyen, Translating tridents, Comm. Partial Differential Equations 34 (2009), no. 13, 257280. MR 2512861 (2010c:53100), http://dx.doi.org/10.1080/03605300902768685
 [SS93]
 H. M. Soner and P. E. Souganidis, Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature, Comm. Partial Differential Equations 18 (1993), no. 56, 859894. MR 1218522 (94c:53011), http://dx.doi.org/10.1080/03605309308820954
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Additional Information
Stephen Kleene
Affiliation:
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Email:
skleene@math.jhu.edu
Niels Martin Møller
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email:
moller@math.mit.edu
DOI:
http://dx.doi.org/10.1090/S000299472014057218
PII:
S 00029947(2014)057218
Received by editor(s):
August 12, 2010
Received by editor(s) in revised form:
September 9, 2011
Published electronically:
March 26, 2014
Article copyright:
© Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
