Abstract
We study the evolution of compact
convex hypersurfaces in
hyperbolic space
Funding source: Australian Research Council
Award Identifier / Grant number: DP1200097
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11271132
Award Identifier / Grant number: 11071212
Award Identifier / Grant number: 11131007
Funding statement: The first author was partly supported by Discovery Projects grant DP1200097 of the Australian Research Council. The second author was partly supported by National Natural Science Foundation of China under grants 11271132, 11071212 and 11131007. The authors are grateful for the hospitality of the Mathematical Sciences Center of Tsinghua University where the research was carried out.
References
[1] R. Alessandroni and C. Sinestrari, Evolution of hypersurfaces by powers of the scalar curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 3, 541–571. 10.2422/2036-2145.2010.3.05Suche in Google Scholar
[2] B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 151–171. 10.1007/BF01191340Suche in Google Scholar
[3] B. Andrews, Contraction of convex hypersurfaces in Riemannian spaces, J. Differential Geom. 39 (1994), no. 2, 407–431. 10.4310/jdg/1214454878Suche in Google Scholar
[4] B. Andrews, Gauss curvature flow: The fate of the rolling stones, Invent. Math. 138 (1999), no. 1, 151–161. 10.1007/s002220050344Suche in Google Scholar
[5] B. Andrews, Positively curved surfaces in the three-sphere, Proceedings of the international congress of mathematicians, ICM 2002. Vol. II: Invited lectures (Beijing 2002), Higher Education Press, Beijing (2002), 221–230. Suche in Google Scholar
[6] B. Andrews, Fully nonlinear parabolic equations in two space variables, preprint (2004), http://arxiv.org/abs/math/0402235. Suche in Google Scholar
[7] B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. reine angew. Math. 608 (2007), 17–33. 10.1515/CRELLE.2007.051Suche in Google Scholar
[8] B. Andrews, Moving surfaces by non-concave curvature functions, Calc. Var. Partial Differential Equations 39 (2010), no. 3–4, 649–657. 10.1007/s00526-010-0329-zSuche in Google Scholar
[9] B. Andrews and C. Baker, Mean curvature flow of pinched submanifolds to spheres, J. Differential Geom. 85 (2010), no. 3, 357–395. 10.4310/jdg/1292940688Suche in Google Scholar
[10] B. Andrews and X. Chen, Surfaces moving by powers of Gauss curvature, Pure Appl. Math. Q. 8 (2012), no. 4, 825–834. 10.4310/PAMQ.2012.v8.n4.a1Suche in Google Scholar
[11] B. Andrews and C. Hopper, The Ricci flow in Riemannian geometry, Lecture Notes in Math. 2011, Springer, Heidelberg 2011. 10.1007/978-3-642-16286-2Suche in Google Scholar
[12] B. Andrews and J. McCoy, Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3427–3447. 10.1090/S0002-9947-2012-05375-XSuche in Google Scholar
[13] C. Baker, The mean curvature flow of submanifolds of high codimension, Ph.D. thesis, The Australian National University, 2010, http://arxiv.org/abs/1104.4409. Suche in Google Scholar
[14] E. Cabezas-Rivas and V. Miquel, Volume preserving mean curvature flow in hyperbolic space, preprint (2006), http://arxiv.org/abs/math/0611216. 10.1512/iumj.2007.56.3060Suche in Google Scholar
[15] B. Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Differential Geom. 22 (1985), no. 1, 117–138. 10.4310/jdg/1214439724Suche in Google Scholar
[16] B. Chow, Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), no. 1, 63–82. 10.1007/BF01389153Suche in Google Scholar
[17] E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. reine angew. Math. 357 (1985), 1–22. 10.1515/crll.1985.357.1Suche in Google Scholar
[18] K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), no. 3, 547–569. 10.1007/BF01232278Suche in Google Scholar
[19] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153–179. 10.4310/jdg/1214440433Suche in Google Scholar
[20] R. S. Hamilton, Convex hypersurfaces with pinched second fundamental form, Commun. Anal. Geom. 2 (1994), no. 1, 167–172. 10.4310/CAG.1994.v2.n1.a10Suche in Google Scholar
[21] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266. 10.4310/jdg/1214438998Suche in Google Scholar
[22] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), no. 3, 463–480. 10.1007/BF01388742Suche in Google Scholar
[23] G. Huisken, Deforming hypersurfaces of the sphere by their mean curvature, Math. Z. 195 (1987), 205–219. 10.1007/BF01166458Suche in Google Scholar
[24] G. Huisken and A. Polden, Geometric evolution equations for hypersurfaces, Calculus of variations and geometric evolution problems. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (CIME) (Cetraro 1996), Lecture Notes Math. 1713, Springer, Berlin (1999), 45–84. 10.1007/BFb0092669Suche in Google Scholar
[25] Q.-R. Li, Surfaces expanding by the power of the Gauss curvature flow, Proc. Amer. Math. Soc. 138 (2010), no. 11, 4089–4102. 10.1090/S0002-9939-2010-10431-8Suche in Google Scholar
[26]
O. C. Schnürer,
Surfaces contracting with speed
[27] O. C. Schnürer, Surfaces expanding by the inverse Gauß curvature flow, J. reine angew. Math. 600 (2006), 117–134. 10.1515/CRELLE.2006.088Suche in Google Scholar
[28] F. Schulze, Convexity estimates for flows by powers of the mean curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), no. 2, 261–277. 10.2422/2036-2145.2006.2.06Suche in Google Scholar
[29] K. Tso, Deforming a hypersurface by its Gauss–Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), no. 6, 867–882. 10.1002/cpa.3160380615Suche in Google Scholar
[30] C. Wu, D. Tian and G. Li, Forced flow by powers of the mth mean curvature, Armen. J. Math. 3 (2010), no. 2, 61–91. Suche in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Un théorème de Lefschetz–Hopf pour les fonctions à itérées compactes
- Curvature flow in hyperbolic spaces
- The mean square of the product of the Riemann zeta-function with Dirichlet polynomials
- Minimality via second variation for microphase separation of diblock copolymer melts
- Analytic function theory for operator-valued free probability
- Counting elliptic curves with prescribed torsion
- E1-degeneration of the irregular Hodge filtration
- Topological invariance of the homological index
- Regularity and Bernstein-type results for nonlocal minimal surfaces
- Regularizing properties of the twisted Kähler–Ricci flow
Artikel in diesem Heft
- Frontmatter
- Un théorème de Lefschetz–Hopf pour les fonctions à itérées compactes
- Curvature flow in hyperbolic spaces
- The mean square of the product of the Riemann zeta-function with Dirichlet polynomials
- Minimality via second variation for microphase separation of diblock copolymer melts
- Analytic function theory for operator-valued free probability
- Counting elliptic curves with prescribed torsion
- E1-degeneration of the irregular Hodge filtration
- Topological invariance of the homological index
- Regularity and Bernstein-type results for nonlocal minimal surfaces
- Regularizing properties of the twisted Kähler–Ricci flow