调和映射与调和复结构
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摘要
本文的研究内容主要分为两个部分.
第一部分是研究调和映射的Liouville型定理在复的情形下的类比.我们主要证明了从n(n≥2)维复平面Cn到任意Kahler流形的具有有限部分能量的调和映射在一定条件下是全纯的.
第二部分是研究作者引进的调和复结构概念,它是介于复结构和Kahler结构之间的一种几何结构.如同极小子流形是全测地子流形的推广一样,调和复结构是Kahler结构的自然推广.我们得到了关于调和复结构的一系列结果.特别地,证明了具有标准度量的S6上不容许任何调和复结构.另外,我们也研究了更一般的A∈r(TM(?)TM*)的迹的一些有趣性质.
The thesis is separated in two parts.
In first part, we study the complex analogousness of Liouville type theorems for harmonic maps. We mainly prove that, under an assumption, a harmonic map of finite (?) energy from Cn (n≥2) to any Kahler manifold must be a holomorphic map.
In the second part, the author introduce a new concept so called harmonic complex structure. It is a new structure intermediates between complex structure and Kahler structure. Harmonic complex structure is the natural generalization of Kahler structure, just as minimal submanifold is the generalization of totally geodesic submanifold. We obtain a series of results about harmonic complex structures. Particularly, we prove that S6 with standard metric can not admit any harmonic complex structure. Moreover, we also get some interesting results of the trace of A∈Γ(TM(?)TM*).
第一部分是研究调和映射的Liouville型定理在复的情形下的类比.我们主要证明了从n(n≥2)维复平面Cn到任意Kahler流形的具有有限部分能量的调和映射在一定条件下是全纯的.
第二部分是研究作者引进的调和复结构概念,它是介于复结构和Kahler结构之间的一种几何结构.如同极小子流形是全测地子流形的推广一样,调和复结构是Kahler结构的自然推广.我们得到了关于调和复结构的一系列结果.特别地,证明了具有标准度量的S6上不容许任何调和复结构.另外,我们也研究了更一般的A∈r(TM(?)TM*)的迹的一些有趣性质.
The thesis is separated in two parts.
In first part, we study the complex analogousness of Liouville type theorems for harmonic maps. We mainly prove that, under an assumption, a harmonic map of finite (?) energy from Cn (n≥2) to any Kahler manifold must be a holomorphic map.
In the second part, the author introduce a new concept so called harmonic complex structure. It is a new structure intermediates between complex structure and Kahler structure. Harmonic complex structure is the natural generalization of Kahler structure, just as minimal submanifold is the generalization of totally geodesic submanifold. We obtain a series of results about harmonic complex structures. Particularly, we prove that S6 with standard metric can not admit any harmonic complex structure. Moreover, we also get some interesting results of the trace of A∈Γ(TM(?)TM*).
引文
[BPV]W.Barth, C.Peters, A.Van de Ven, Compact Complex Surfaces, Ergeb-nisse der Math., Band 4, Springer-Verlag,1984.
[G]A. Gray Nearly Kdhler manifolds, J. Differential Geometry,4,283-309, 1970.
[GRSB]W.D.Garber, SNM.Ruijsenaass, E.Seider and D.Burns, On finite action solution of nonlinear σ-model, Ann. Phy.,119,1979.
[H]S.Hildebrandt, Liouville theorems for harmonic mappings, and an ap-proach to Bernstein theorems, seminar on differential geometry,107-132, Edited by S.T.Yau, Ann. of math study 102, Princeton,1982.
[KN]S.Kobayashi and K.Nomizu, Foundations of differntial geometry Vol 2, Wiley, New York,1969.
[L]C.LeBrun, Orthogonal complex structure on S6, Proc. Amer. Math. Soc. 101:136-138,1987.
[S]H.C.J.Sealey, Some conditions ensuring the vanishing of harmonic dif-feretial forms with applications to harmonic maps and Yang-Mills theory, Math. Proc.Camb.Phil.Soc.,91:441-452,1982.
[SY]R.Schoen and S.T.Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds with nonnegative Ricci curvature, Comment Math Helv,39,333-341,1976.
[SiY]Y.T.Siu and S.T.Yau, Compact Kdhler manifolds of positive bisectional curvature, Invention Math,59:189-204,1980.
[Wal]万建明,调和复结构,数学年刊A缉,30卷,第六期,761-764,2009.
[Wa2]Jianming Wan, Harmonic maps from Cn to Kdhler manifolds, sub-mitted.
[W]C.M.Wood, Harmonic almost-complex structures, Compositoi Mathe-matica,99:183-212,1995.
[WC]伍鸿熙,陈维桓,黎曼几何选讲,北京大学出版社,1993.
[X]忻元龙,调和映照,上海科学技术出版社,1995.
[Y]S.T.Yau, Parallelizable manifolds without complex structures, Topology, 15:51-53,1976.
[YS]丘成桐,孙理察,微分几何讲义,高等教育出版社,2004.
[G]A. Gray Nearly Kdhler manifolds, J. Differential Geometry,4,283-309, 1970.
[GRSB]W.D.Garber, SNM.Ruijsenaass, E.Seider and D.Burns, On finite action solution of nonlinear σ-model, Ann. Phy.,119,1979.
[H]S.Hildebrandt, Liouville theorems for harmonic mappings, and an ap-proach to Bernstein theorems, seminar on differential geometry,107-132, Edited by S.T.Yau, Ann. of math study 102, Princeton,1982.
[KN]S.Kobayashi and K.Nomizu, Foundations of differntial geometry Vol 2, Wiley, New York,1969.
[L]C.LeBrun, Orthogonal complex structure on S6, Proc. Amer. Math. Soc. 101:136-138,1987.
[S]H.C.J.Sealey, Some conditions ensuring the vanishing of harmonic dif-feretial forms with applications to harmonic maps and Yang-Mills theory, Math. Proc.Camb.Phil.Soc.,91:441-452,1982.
[SY]R.Schoen and S.T.Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds with nonnegative Ricci curvature, Comment Math Helv,39,333-341,1976.
[SiY]Y.T.Siu and S.T.Yau, Compact Kdhler manifolds of positive bisectional curvature, Invention Math,59:189-204,1980.
[Wal]万建明,调和复结构,数学年刊A缉,30卷,第六期,761-764,2009.
[Wa2]Jianming Wan, Harmonic maps from Cn to Kdhler manifolds, sub-mitted.
[W]C.M.Wood, Harmonic almost-complex structures, Compositoi Mathe-matica,99:183-212,1995.
[WC]伍鸿熙,陈维桓,黎曼几何选讲,北京大学出版社,1993.
[X]忻元龙,调和映照,上海科学技术出版社,1995.
[Y]S.T.Yau, Parallelizable manifolds without complex structures, Topology, 15:51-53,1976.
[YS]丘成桐,孙理察,微分几何讲义,高等教育出版社,2004.