子流形的几何刚性定理和微分球面定理
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摘要
本文着重研究子流形的几何与拓扑的若干问题,获得了球面中平行平均曲率子流形的刚性定理,完备子流形的微分球面定理,局部共形平坦流形上Yamabe流的收敛性定理等结果.本文主要由三部分(第二至第四章)组成.
本文第一部分证明了球面中具有平行平均曲率子流形的若干外蕴刚性定理.1986年,H. Gauchman[G]获得了关于球面中紧致极小子流形的著名刚性定理：设Mn是n+p维单位球面Sn+p中n维紧致极小子流形,h为M的第二基本形式,若对任意单位切向量u∈UM,有σ(u)≤1/3,其中则σ(u)（？）0,即M是全测地子流形；或者σ(u)（？）1/3,并且可完全确定这类子流形的几何结构.在本文第二章中,我们对Gauchman的定理作了如下推广：设Mn是n+p维单位球面Sn+p(1)中n维完备平行平均曲率子流形,H为其平均曲率,h为无迹(化)第二基本形式,若对任意单位切向量u∈UM,有则σ(u)三0,即M是全脐子流形；或者σ(u)三1/3,并且可完全确定M的几何结构.我们进一步证明了下述刚性定理：设Mn是Sn+p(1)中n维紧致的平行平均曲率子流形,h为M的第二基本形式,若对任意单位切向量u,υ∈UM,有则M是全脐球面
本文第二部分研究了完备子流形的微分球面定理.流形的曲率与拓扑是整体微分几何中的核心课题之一.Andersen、Berger、Brendle、Cheeger、Chenrn、Colding、Gromoll、Gromov、Grove、Hamilton、Klingenberg、Perelman、Schoen、Shiohama、Yau等一批著名学者对这一领域作出了重要的贡献.最近,H.W.Xu、E. T. Zhao和J. R. Gu[XZl,XG]获得了数量曲率拼挤条件下常曲率空间形式中完备子流形的最佳微分球面定理.在第三章中,我们运用S. Brendle的Ricci流收敛性定理,证明了下述结果：如果Mn为单位球面Sn+p(1)中n维完备子流形,且对任意单位切向量u∈UM,有σ(u)<1/3,那么Mn必微分同胚于n维标准球面Sn(1).我们还证明了截面曲率拼挤条件下子流形的微分球面定理.
本文第三部分研究了一类局部共形平坦黎曼流形上Yamabe流的收敛性问题.B. Chow[Ch]曾证明：若M为具有正Ricci曲率的紧致局部共形平坦流形,则M上正规化Yamabe流的解在C∞范数的意义下收敛到一个常曲率度量.最近,H. W. Xu和E. T. Zhao[XZ2]证明了下述刚性定理：若M为具有常数量曲率的紧致局部共形平坦流形,且其无迹(化)Ricci曲率张量的Ln/2范数小于某一正常数,则M等距于常曲率空间形式.受上述结果的启发,我们证明了下述结果：若M为具有正数量曲率的紧致局部共形平坦流形,其无迹Ricci曲率张量的Lq范数(q>2)小于某一正常数,则正规化Yamabe流的解在C∞范数的意义下收敛到一个常曲率度量.由此得到了关于局部共形平坦黎曼流形的一个微分球面定理.
In this paper, we mainly study several problems on geometry and topology of submanifolds. We obtain rigidity theorems for submanifolds with parallel mean curvature in the unit sphere, differentiable sphere theorems of complete submanifolds, a convergence theorem of yamabe flow on locally conformally flat manifolds, etc. This paper consists of three parts (Chapter 2 to Chapter 4).
In Chapter 2, we prove some extrinsic rigidity theorems of submanifolds with parallel mean curvature. In 1986, H. Gauchman[G] proved a famous rigidity the-orem for compact minimal submanifolds in a sphere:Let M be an n-dimensional compact minimal submanifold in Sn+p(1), h be the second fundamental form, ifσ(u)≤1/3 holds for any unit vector u∈UM, where then eitherσ(u)≡0, i.e. M is totally geodesic; orσ(u)≡1/3. Moreover, all minimal immersions satisfyingσ(u)≡1/3 can be determined. In this paper, we generalized Gauchman's theorem as follows:Let M be an n-dimensional complete subman-ifold with parallel mean curvature in the unit sphere Sn+p(1), H be the mean curvature, h be the traceless second fundamental form, if for any unit tangent vector u∈UM, then eitherσ(u)≡0 and M is a totally umbilical sphere; orσ(u)=1/3 and the geometric classification of such submanifold M is given. We also prove a rigidity theorem as follows:Let M be a compact submanifold of unit sphere Sn+p(1) with parallel mean curva-ture, if for any unit tangent vector u,υ∈UM, then M is a totally umbilical sphere
In Chapter 3, we prove differentiable sphere theorems for Riemannian sub-manifolds. The study of curvature and topology of manifolds is one of the core subjects in global differentiable geometry. Andersen, Berger, Brendle, Cheeger, Chern, Colding, Gromoll, Gromov, Grove, Hamilton, Klingenberg, Perelman, Schoen, Shiohama, Yau etc.,have made great contributions to this subject. In this thesis, using the convergence theorem of Ricci flow proved by S. Brendle, we prove the following differentiable sphere theorem:Let M be an n-dimensional complete submanifold of Sn+p(1), andσ(u)<1/3 for any unit tangent vector u∈UM, then Mn is diffeomorphic to Sn(1). We also prove a differentiable sphere theorem for submanifolds with pinched sectional curvature.
In Chapter 4, we study the convergence of yamabe flow on locally confor-mally flat manifolds with integral pinching condition. B. Chow [Chow] proved that:Let M be a compact locally conformally flat manifolds with positive Ricci curvature, then the solution of normalized Yamabe flow on M converges in C∞-norm to a constant curvature metric. Recently, H. W. Xu and E. T. Zhao [XZ1] proved that:If M is a locally conformally flat manifolds with constant scalar curvature, and if the Ln/2 norm of traceless Ricci curvature tensor is less than a positive constant, then M is isometric to a space form. Inspired by the results above, we prove the following result:Let M be a compact locally conformally flat manifolds with positive scalar curvature, and if the Lq-norm(q> 2) of the trace-free Ricci curvature is bounded from above by a positive constant, then the solution of the normalized Yamabe flow on M converges to a constant curvature metric. As a consequence, we obtain a differentiable sphere theorem for locally conformally flat manifolds.
本文第一部分证明了球面中具有平行平均曲率子流形的若干外蕴刚性定理.1986年,H. Gauchman[G]获得了关于球面中紧致极小子流形的著名刚性定理：设Mn是n+p维单位球面Sn+p中n维紧致极小子流形,h为M的第二基本形式,若对任意单位切向量u∈UM,有σ(u)≤1/3,其中则σ(u)（？）0,即M是全测地子流形；或者σ(u)（？）1/3,并且可完全确定这类子流形的几何结构.在本文第二章中,我们对Gauchman的定理作了如下推广：设Mn是n+p维单位球面Sn+p(1)中n维完备平行平均曲率子流形,H为其平均曲率,h为无迹(化)第二基本形式,若对任意单位切向量u∈UM,有则σ(u)三0,即M是全脐子流形；或者σ(u)三1/3,并且可完全确定M的几何结构.我们进一步证明了下述刚性定理：设Mn是Sn+p(1)中n维紧致的平行平均曲率子流形,h为M的第二基本形式,若对任意单位切向量u,υ∈UM,有则M是全脐球面
本文第二部分研究了完备子流形的微分球面定理.流形的曲率与拓扑是整体微分几何中的核心课题之一.Andersen、Berger、Brendle、Cheeger、Chenrn、Colding、Gromoll、Gromov、Grove、Hamilton、Klingenberg、Perelman、Schoen、Shiohama、Yau等一批著名学者对这一领域作出了重要的贡献.最近,H.W.Xu、E. T. Zhao和J. R. Gu[XZl,XG]获得了数量曲率拼挤条件下常曲率空间形式中完备子流形的最佳微分球面定理.在第三章中,我们运用S. Brendle的Ricci流收敛性定理,证明了下述结果：如果Mn为单位球面Sn+p(1)中n维完备子流形,且对任意单位切向量u∈UM,有σ(u)<1/3,那么Mn必微分同胚于n维标准球面Sn(1).我们还证明了截面曲率拼挤条件下子流形的微分球面定理.
本文第三部分研究了一类局部共形平坦黎曼流形上Yamabe流的收敛性问题.B. Chow[Ch]曾证明：若M为具有正Ricci曲率的紧致局部共形平坦流形,则M上正规化Yamabe流的解在C∞范数的意义下收敛到一个常曲率度量.最近,H. W. Xu和E. T. Zhao[XZ2]证明了下述刚性定理：若M为具有常数量曲率的紧致局部共形平坦流形,且其无迹(化)Ricci曲率张量的Ln/2范数小于某一正常数,则M等距于常曲率空间形式.受上述结果的启发,我们证明了下述结果：若M为具有正数量曲率的紧致局部共形平坦流形,其无迹Ricci曲率张量的Lq范数(q>2)小于某一正常数,则正规化Yamabe流的解在C∞范数的意义下收敛到一个常曲率度量.由此得到了关于局部共形平坦黎曼流形的一个微分球面定理.
In this paper, we mainly study several problems on geometry and topology of submanifolds. We obtain rigidity theorems for submanifolds with parallel mean curvature in the unit sphere, differentiable sphere theorems of complete submanifolds, a convergence theorem of yamabe flow on locally conformally flat manifolds, etc. This paper consists of three parts (Chapter 2 to Chapter 4).
In Chapter 2, we prove some extrinsic rigidity theorems of submanifolds with parallel mean curvature. In 1986, H. Gauchman[G] proved a famous rigidity the-orem for compact minimal submanifolds in a sphere:Let M be an n-dimensional compact minimal submanifold in Sn+p(1), h be the second fundamental form, ifσ(u)≤1/3 holds for any unit vector u∈UM, where then eitherσ(u)≡0, i.e. M is totally geodesic; orσ(u)≡1/3. Moreover, all minimal immersions satisfyingσ(u)≡1/3 can be determined. In this paper, we generalized Gauchman's theorem as follows:Let M be an n-dimensional complete subman-ifold with parallel mean curvature in the unit sphere Sn+p(1), H be the mean curvature, h be the traceless second fundamental form, if for any unit tangent vector u∈UM, then eitherσ(u)≡0 and M is a totally umbilical sphere; orσ(u)=1/3 and the geometric classification of such submanifold M is given. We also prove a rigidity theorem as follows:Let M be a compact submanifold of unit sphere Sn+p(1) with parallel mean curva-ture, if for any unit tangent vector u,υ∈UM, then M is a totally umbilical sphere
In Chapter 3, we prove differentiable sphere theorems for Riemannian sub-manifolds. The study of curvature and topology of manifolds is one of the core subjects in global differentiable geometry. Andersen, Berger, Brendle, Cheeger, Chern, Colding, Gromoll, Gromov, Grove, Hamilton, Klingenberg, Perelman, Schoen, Shiohama, Yau etc.,have made great contributions to this subject. In this thesis, using the convergence theorem of Ricci flow proved by S. Brendle, we prove the following differentiable sphere theorem:Let M be an n-dimensional complete submanifold of Sn+p(1), andσ(u)<1/3 for any unit tangent vector u∈UM, then Mn is diffeomorphic to Sn(1). We also prove a differentiable sphere theorem for submanifolds with pinched sectional curvature.
In Chapter 4, we study the convergence of yamabe flow on locally confor-mally flat manifolds with integral pinching condition. B. Chow [Chow] proved that:Let M be a compact locally conformally flat manifolds with positive Ricci curvature, then the solution of normalized Yamabe flow on M converges in C∞-norm to a constant curvature metric. Recently, H. W. Xu and E. T. Zhao [XZ1] proved that:If M is a locally conformally flat manifolds with constant scalar curvature, and if the Ln/2 norm of traceless Ricci curvature tensor is less than a positive constant, then M is isometric to a space form. Inspired by the results above, we prove the following result:Let M be a compact locally conformally flat manifolds with positive scalar curvature, and if the Lq-norm(q> 2) of the trace-free Ricci curvature is bounded from above by a positive constant, then the solution of the normalized Yamabe flow on M converges to a constant curvature metric. As a consequence, we obtain a differentiable sphere theorem for locally conformally flat manifolds.
引文
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[BS1]S. Brendle and R. Schoen, Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc.,22(2009),287-307.
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[Hu3]G. Huisken, Deforming hypersurfaces of the sphere by their mean curva-ture, Math. Z.,195(1987),205-219.
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[Be]M. Berger, Les varietes Riemanniennes (1/4)-pincees, Ann. Scuola Norm. Sup. Pisa,14(1960),161-170.
[Bo]R. Bott, Lectures on Morse theory, old and new. Bull. Amer. Math. Soc (new series),7(1982),331-358.
[Bre]S. Brendle, A general convergence result for the Ricci flow in higher dimen-sions, Duke Math. J.,145(2008),585-601.
[BS1]S. Brendle and R. Schoen, Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc.,22(2009),287-307.
[BS2]S. Brendle and R. Schoen, Sphere theorems in geometry, arXiv:0904.2604, 2009.
[BW]C. Bohm and B. Wilking, Manifolds with positive curvature operators are space forms, Ann. of Math.,167(2008),1079-1097.
[C]Q. Chen, A pinching theorem for submanifolds of unit spheres. Acta Math. Sinica (Chin. Ser.),39(1996),57-63.
[Ch]B. Chow, The Yamabe flow on locally conformally flat manifolds with pos-itive ricci curvature, Comm. Pure Appl. Math.,45(1992),1003-1014.
[Chen]B. Y. Chen, Some pinching and classification theorems for minimal sub-manifolds, Arch. Math.,60(1993),568-578.
[CDK]S. S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, In:Functional analysis and related fields, pp.59-75. Berlin, Heidelberg, New york:Springer 1970.
[CLN]B. Chow, P. Lu and L. Ni, Hamilton's Ricci flow, Graduate Studies in Mathematics 77, Science Press, New York.
[CN]Q. M. Cheng and H. Nakagawa, Totally umbilic hypersurfaces, Hiroshima Math. J.,20(1990),1-10.
[CZ]H. D. Cao and X. P. Zhu, A complete proof of the Poincare and geometriza-tion conjectures-Application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math.,10(2006),165-492.
[G]H. Gauchman, Minimal submanifolds of a sphere with bounded second fun-damental form, Trans. Amer. Math. Soc.,298(1986),779-791.
[GC]D. Gromoll, Diffrenzierbare Strukturen und Metriken positive Krummung auf Spharen, Math. Ann.,164(1966),353-371.
[GS]K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math., 106(1977),201-211.
[Hal]R. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom.,17(1982),255-306.
[Ha2]R. Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom.,5(1997),1-92.
[Ha3]R. Hamilton, The Ricci flow on surfaces, Math. General. Relaticity, Con-temp. Math.,71(1988),237-261.
[He]E. Hebey, Nonlinear Analysis on Manifolds:Sobolev Spaces and Inequali-ties, Courant Institute of Mathematical Science, Vol.5,2000.
[Hul]G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Diff. Geom.,20(1984),237-266.
[Hu2]G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math.,84(1986),463-480.
[Hu3]G. Huisken, Deforming hypersurfaces of the sphere by their mean curva-ture, Math. Z.,195(1987),205-219.
[K]W. Klingenberg, Uber Riemannsche Mannigfaltigkeiten mit positiver Krummung, Comment. Math. Helv.,35(1961),47-54.
[La]B. Lawson, Local rigidity theorems for minimal hypersurfaces, Ann. of Math.,89(1969),179-185.
[Le1]P. F. Leung, On a relation between the topology and the intrinsic and extrinsic geometries of a compact submanifold, Proc. Edindurgh Math. Soc., 28(1985),305-311.
[Le2]P. F. Leung, On the topology of a compact submanifold of a sphere with bounded second fundamental form, Manus. Math.,79(1993),183-185.
[LL]A. M. Li and J. M. Li, An intrinsic rigidity theorem for minimal submani-folds in a sphere, Arch. Math.,58(1992),582-594.
[LS]H. Lawson and J. Simons, On stable currents and their application to global problems in real and complex geometry, Ann. of Math.,98(1973),427-450.
[M]J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure. Appl. Math.,17(1964),101-134.
[Ok1]M. Okumura, Submanifolds and a pinching problem on the second funda-mental tensors, Trans. Amer. Math. Soc.,178(1973),285-291.
[Ok2]M. Okumura, Hypersurfaces and a pinching problem on the second fun-damental tensor, Amer. J. Math.,96(1974),207-213.
[PRS]S. Pigola, M. Rigoli and A. G. Setti, Some characterizations of space-forms, Tran. Amer. Math. Soc.,359(2007),1817-1828.
[Ra]H.E. Rauch, A contribution to differential geometry in the large, Ann. of Math.,54(1951),38-55.
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