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Riemann-Finsler Geometry with Applications to Information Geometry

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Chinese Annals of Mathematics,Series B 2006年第1期27卷 73-94页

作者： Zhongmin shen Dedicated to the memory of Shiing-shen Chern

Information geometry is a new branch in mathematics, originated from the applications of differential geometry to statistics. In this paper we briefly introduce Riemann-Finsler geometry, by which we establish Information Geometry on a much broader base, so that the potential applications of Information Geometry will be beyond statistics.

关键词： Riemann-Finsler geometry Information geometry

On the Kahler-Ricci Flow on Projective Manifolds of General Type

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Chinese Annals of Mathematics,Series B 2006年第2期27卷 179-192页

作者： Gang TIAN Zhou ZHANG Dedicated to the memory of Shiing-shen Chern

This note concerns the global existence and convergence of the solution for Kahler-Ricci flow equation when the canonical class, Kx, is numerically effective and big. We clarify some known results regarding this flow ... 详细信息

关键词： Geometric evolution equations Minimal model program

Geometry of Ricci Solitons

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Chinese Annals of Mathematics,Series B 2006年第2期27卷 121-142页

作者： Huai-Dong CAO Dedicated to the memory of Shiing-shen Chern

Ricci solitons are natural generalizations of Einstein metrics on one hand, and are special solutions of the Ricci flow of Hamilton on the other hand. In this paper we survey some of the recent developments on Ricci s... 详细信息

关键词： Ricci soliton Singularity of Ricci flow Stability Gaussian density

A Geometric Problem and the Hopf Lemma. II

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Chinese Annals of Mathematics,Series B 2006年第2期27卷 193-218页

作者： Louis NIRENBERG Dedicated to the memory of Shiing-shen Chern

A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in R^n＋1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn＋1 =constant in case M satisfies： for any two points （X^1, ^↑Xn＋1）, （X^1, Xn＋1） on M, with Xn＋1 〉^↑Xn＋1, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part I dealt with corresponding one dimensional problems.

关键词： Hopf Lemma Maximum principle Moving planes Symmetry Mean curvature

Associative Cones and Integrable Systems

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Chinese Annals of Mathematics,Series B 2006年第2期27卷 153-168页

作者： Chuu-Lian TERNG Dedicated to the memory of Shiing-shen Chern

Abstract We identify R^7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S^6. It is known that a cone over a surface M in S^6 is an associative submanifold of R^7 if and only if M is almost complex in S^6. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S^6 are the equation for primitive maps associated to the 6-symmetric space G2/T^2, and use this to explain some of the known results. Moreover, the equation for S^1-symmetric almost complex curves in S^6 is the periodic Toda lattice, and a discussion of periodic solutions is given.

关键词： Octonions Associative cone Almost complex curve Primitive map

A priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry

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Chinese Annals of Mathematics,Series B 2006年第2期27卷 169-178页

作者： Xu-Jia WANG Dedicated to the memory of Shiing-shen Chern

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations *** the manifold is locally conformally flat or the Ricci curvature is positive.

关键词： Conformal geometry Elliptic equation Ricci curvature

Ricci Curvature and Fundamental Group

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Chinese Annals of Mathematics,Series B 2006年第2期27卷 113-120页

作者： Yuanlong XIN Dedicated to the memory of Shiing-shen Chern

Abstract By refined volume estimates in terms of Ricci curvature, the two results due to J. Milnor （1968） are generalized.

关键词： Ricci curvature Fundamental group Volume growth

η-Invariant and Flat Vector Bundles

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Chinese Annals of Mathematics,Series B 2006年第1期27卷 67-72页

作者： Xiaonan MA Weiping ZHANG Dedicated to the memory of Shiing-shen Chern

We present an alternate definition of the mod Z component of the Atiyah-Patodi-Singer η invariant associated to（not necessary unitary ）flat vector bundles,which identifies explicitly its realandimaginary *** is done by combining a deformation of flatconnections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah Patodi and Singer.

关键词： Flat vector bundle η-Invariant ρ-Invariant

The Einstein-Scalar Field Constraints on Asymptotically Euclidean Manifolds

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Chinese Annals of Mathematics,Series B 2006年第1期27卷 31-52页

作者： Yvonne CHOQUET-BRUHAT James ISENBERG Daniel POLLACK Dedicated to the memory of Shiing-shen Chern

By using the conformal method, solutions of the Einstein-scalar ficld gravitational constraint equations are obtained. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang-Mills fields, because the scalar field introduces three extra terms into the Lichnerowicz equation, rather than just one. The proofs are constructive and allow for arbitrary dimension （〉 2） as well as low regularity initial data.

关键词： Einstein-scalar field Constraints Asymptotically Euclidian