A Bridge Principle for Harmonic Maps

We prove a bridge principle for harmonic maps between generalmanifolds.     

Harmonic Diffeomorphisms between Complete Surfaces

We prove an existing theorem of homotopic harmonic diffeomorphisms between complete surfaces of finite total curvature and nontrivial genus. This is a generalization of a theorem of Jost and Schoen [4].     

Growth Sequences for Circle Diffeomorphisms

We obtain results on the growth sequences of the differential for iterations of circle diffeomorphisms without periodic points. Received: June 2005 Revision: December 2005 Accepted: February 2006     

Geometry of Infinitesimal Harmonic Transformations

One of the present authors defined the infinitesimal harmonictransformation in a Riemannian manifold. This paperis devoted to the study of the local and global geometryinfinitesimal harmonic transformations.     

极小曲面中的若干问题

本文综述了Ｅ＾ｎ和Ｓ＾ｎ中极小曲面的若干经典结果和最新发展，指出了一些尚未解决的问题，在第４节中，对Ｅ＾３中极小曲面的Ｆｕｊｉｍｏｔｏ定理给出了一个更直接的证明。     

A Theorem of Liouville Type for Harmonic Morphisms

Let : M N be a harmonic morphism from a complete noncompact Riemannian manifold M with nonnegative Ricci curvature to a complete Riemannian manifold N with nonpositive scalar curvature. We show if the energy of is finite, then is constant. This can be compared with a similar result for harmonic maps when N has nonpositive sectional curvature due to Schoen and Yau.     

Minimal Surfaces in a Cone

We prove the convex hull property for properly immersed minimal hypersurfaces in a cone of ⁿ . We deal with the existence of new barriers for the maximum principle application in noncompact truncated tetrahedral domains of ³, describing the space of such domainsadmitting barriers of this kind. Nonexistence results for nonflatminimal surfaces whose boundary lies in opposite faces of a tetrahedraldomain are obtained. Finally, new simple closed subsets of ³ whichhave the property of intersecting any properly immersed minimal surfaceare shown.     

整体Pinching引理及其在子流形的几何、调和映射和Yang-Mills场中的应用

本文建立了两个解析引理，并将它们应用于予流形的几何、调和映射以及Yang—Mills场理论，从而获得了关于这些对象的几个整体Pinching结果。     

The Energy Growth Property for Harmonic Maps

Ｔｈｅ　Ｅｎｅｒｇｙ　Ｇｒｏｗｔｈ　Ｐｒｏｐｅｒｔｙ　ｆｏｒ　Ｈａｒｍｏｎｉｃ　ＭａｐｓＴｈｅＥｎｅｒｇｙＧｒｏｗｔｈＰｒｏｐｅｒｔｙｆｏｒＨａｒｍｏｎｉｃＭａｐｓ￥ＬｉＨｏｎｇｙｉ；ＺｈａｏＤｉ（ＮｏｒｔｈＣｈｉｎａｉｎｓｔｉｔｕｔｅｏｆＷａｔｅｒ...     

Lifts of Diffeomorphisms of Orbit Spaces for Representations of Compact Lie Groups

A structure of the group of diffeomorphisms for the orbit space of an orthogonal representation G> O(V) of a compact Lie group G is studied. For a finite G, it is proved that each diffeomorphism of the orbit space V/G has a smooth lift to V and the component group of the group of Diff(V/G is described, especially in the case when G is a finite Coxeter group. Similar results are obtained for the isotropy representations of some symmetric spaces.     

Conditions for the Existence of SBR Measures for ``Almost Anosov' Diffeomorphisms

A diffeomorphism of a compact manifold is called ``almost Anosov' if it is uniformly hyperbolic away from a finite set of points. We show that under some nondegeneracy condition, every almost Anosov diffeomorphism admits an invariant measure that has absolutely continuous conditional measures on unstable manifolds. The measure is either finite or infinite, and is called SBR measure or infinite SBR measure respectively. Therefore, tends to either an SBR measure or for almost every with respect to Lebesgue measure. ( is the Dirac measure at .) For each case, we give sufficient conditions by using coefficients of the third order terms in the Taylor expansion of at .

     

The Barrier Principle for Minimal Submanifolds of Arbitrary Codimension

We establish a barrier principle for minimal submanifolds of a Riemannianmanifold of arbitrary codimension. We construct examples of barriers fortwo-dimensional minimal surfaces in ⁿ, n 4, and apply these to deduceexistence as well as nonexistence theorems for Plateau's problem.     

A Maximum Principle at Infinity for Surfaces with Constant Mean Curvature in Euclidean Space

Maximum principles at infinity generalize Hopf's maximum principle for hypersurfaces with constant mean curvature in R ⁿ . We establish such a maximum principle for parabolic surfaces in R³ with nonzero constant mean curvature and bounded Gaussian curvature.     

Timelike Minimal Surfaces via Loop Groups

This work consists of two parts. In Part I, we shall give a systematic study of Lorentz conformal structure from structural viewpoints. We study manifolds with split-complex structure. We apply general results on split-complex structure for the study of Lorentz surfaces.In Part II, we study the conformal realization of Lorentz surfaces in the Minkowski 3-space via conformal minimal immersions. We apply loop group theoretic Weierstrass-type representation of timelike constant mean curvature for timelike minimal surfaces. Classical integral representation formula for timelike minimal surfaces will be recovered from loop group theoretic viewpoint.     

曲面调和映照的广义Frenet公式及其应用

本文建立了从曲面到复Grassmamn流形调和映照的广义Frenet公式。作为应用，我们得到了调和映照为强共形的一个等价条件。我们也讨论了等距调和映照的曲率pinching性质。从而改进了有关伪全纯曲线的相应结果。     

曲面到U（N）中有限的调和映射

本文证明了如果ｓ是从曲面Ω到Ｕ（Ｎ）中有限的调和映射，则Ａｚ，Ａｚ－是幂零的，并给出其逆不成立的反例。     

G2,n中的可兼共形调和曲面

本文通过建立复Ｇｒａｓｓｍａｎｎ流形Ｇ２，ｎ中调和曲面的Ｇａｕｓｓ丛的基本公式，在适当的拓扑条件下给出了Ｇ２，ｎ中可兼共形调和曲面的构造定理．推广和改进了Ｂｕｒｓｔａｌｌ和Ｗｏｏｄ的低亏格曲面的结果．     

A Remark on the Intrinsic Rigidity of Compact Minimal Submanifolds in A Sphere

ＡＲｅｍａｒｋｏｎｔｈｅＩｎｔｒｉｎｓｉｃＲｉｇｉｄｉｔｙｏｆＣｏｍｐａｃｔＭｉｎｉｍａｌＳｕｂｍａｎｉｆｏｌｄｓｉｎＡＳｐｈｅｒｅ￥ＳｈｕＳｈｉｃｈａｎｇ（舒世昌）（ｉａＸｉｎｇｑｉｎ（贾兴琴）（ＸｉａｎｙａｎｇＴｅａｃｈｅｒｓ＇Ｃｏｌｌｅｇｅ，７...     

Evolution Equations for Special Lagrangian 3-Folds in C3

This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C ^m . The previous paper (Math. Ann. 320 (2001), 757–797), defined the idea of evolution data, which includes an (m – 1)-submanifold P in R ⁿ , and constructed a family of special Lagrangian m-folds N in C ^m , which are swept out by the image of P under a 1-parameter family of affine maps _t : R ⁿ C ^m , satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C³. We find a one-to-one correspondence between sets of evolution data with m = 3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C³.Our main results are a number of new families of special Lagrangian 3-foldsin C³, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R³ or ¹× R². Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind.We hope these 3-folds will be helpful in understanding singularities ofcompact special Lagrangian 3-folds in Calabi–Yau 3-folds. This will beimportant in resolving the SYZ conjecture in Mirror Symmetry.     

单叶调和映射

由于在极小曲面理论中的作用，对调和映射的研究已有较长时间。１９８４年以来，经典解析单叶映射的理论被推广至调和单叶映射，并获得许多结论。这些工作引起人们对它的浓厚兴趣。该文介绍这一课题某些重要成果的概貌，并指出一些尚未解决的问题。它共分六个部分：映射定理；单叶调和函数的数值估计；特殊映射；变分方法；境界性质和在极小曲面中的应用。