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Topology and Curvature of Isoparametric Families in Spheres

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Communications in Mathematics and Statistics 2023年第2期11卷 439-475页

作者： Chao Qian Zizhou Tang Wenjiao Yan School of Mathematics and Statistics Beijing Institute of TechnologyBeijing 100081People’s Republic of China Chern Institute of Mathematics and LPMC Nankai UniversityTianjin 300071People’s Republic of China School of Mathematical Sciences Laboratory of Mathematics and Complex SystemsBeijing Normal UniversityBeijing 100875People’s Republic of China

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds.The present paper has two parts.The first part investigates topology of the isoparametric families,namely the homotopy,homeomorphism,or diffeomorphism types,parallelizability,as well as the Lusternik-Schnirelmann category.This part extends substantially the results of Wang(J Differ Geom 27:55-66,1988).The second part is concerned with their curvatures;more precisely,we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

关键词： Isoparametric hypersurface Focal submanifold Homotopy equivalent Homeomorphism diffeomorphism Parallelizability Lusternik-Schnirelmann category Sectional curvature Ricci curvature

diffeomorphisms with the M0-shadowing Property

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Acta Mathematica Sinica,English Series 2019年第11期35卷 1760-1770页

作者： Xu ZHANG Xin Xing WU Department of Mathematics Shandong UniversityWeihai 264209P.R.China School of Sciences Southwest Petroleum UniversityChengdu 610500P.R.China Department of Electronic Engineering City University of Hong KongHong Kong SARP.R.China

This paper studies the M0-shadowing property for the dynamics of diffeomorphisms defined on closed manifolds. The C1 interior of the set of all two dimensional diffeomorphisms with the M0-shadowing property is described by the set of all Anosov diffeomorphisms. The C1-stably M0-shadowing property on a non-trivial transitive set implies the diffeomorphism has a dominated splitting.

关键词： Anosov diffeomorphism average shadowing property Mα-shadowing

EXCESS FUNCTIONS OF RAYS ON COMPLETE NONCOMPACT MANIFOLDS

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Acta Mathematica Scientia 2003年第3期23卷 339-344页

作者：徐森林徐栩 Department of Mathematics Central China Normal UniversityWuhan 430079China School of Mathematics and Statistics Wuhan UniversityWuhan 430072China

This paper gives an estimate of excess functions of rays on complete non-compact manifolds. By using this estimation, the authors can get the results in [3] as corollaries, which asserts that a complete manifold is diffeomorphic to Rn under some curvature and pinching conditions. At last, they obtain a refinement of them with extra Ricci condition.

关键词： Excess functions rays diffeomorphism

Global smooth solution of multi-dimensional non-homogeneous conservation laws

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Progress in Natural Science:Materials International 2004年第10期14卷 15-22页

作者： YANG Xiaozhou 1 and ZHANG Tong 2 (1.Department of Mathematics,Shantou University,Shantou515063,China 2.Institute of Mathematics,Chinese Academy of Sci-ences,Beijing100080,China) Dept. of Math. Shantou Univ. Shantou 515063 China Inst. of Math. Chinese Acad. of Sci. Beijing 100080 China

The initial value problem of multi-dimensional non-homogeneous conservation laws is studied and a sufficient condition of the existence of its global smooth solution is obtained.Additionally,three equivalent propositions of the existence of global smooth solution for multi-dimensional homogeneous scalar conservation laws are also obtained.The global smooth solution is uniquely expressed by the global implicit function that is determined by a scalar equation,not by a system as usual methods do.

关键词： conservation laws global smooth solution global implicit function diffeomorphism

Kobayashi’s and Teichmuller’s Metrics and Bers Complex Manifold Structure on Circle diffeomorphisms

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Acta Mathematica Sinica,English Series 2020年第3期36卷 245-272页

作者： Yun Ping JIANG Department of Mathematics Queens College of the City University of New York FlushingNY 11367-1597USA Department of Mathematics Graduate School of the City University of New York365 Fifth A venueNew YorkNY 10016USA

Given a modulus of continuity ω,we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius transformations preserving the unit disk.We study several distortion properties for diffeomorphisms and quasisymmetric homeomorphisms.Using these distortion properties,we give the Bers complex manifold structure on the Teichm(u| ")ller space TC^1+H as the union of over all0 diffeomorphisms on which we can assign such a structure.Furthermore,we prove that with the Bers complex manifold structure on TC^1+H ,Kobayashi’s metric and Teichmuller’s metric coincide.

关键词： Bers complex manifold structure circle diffeomorphism modulus of continuity quasisymmetric circle homeomorphism Teichmuller space Kobayashi's metric Teichmuller's metric

Two Characterizations of diffeomorphisms of Euclidean Space

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Acta Mathematica Sinica,English Series 2003年第4期19卷 739-744页

作者： LiJunYANG XianWuZENG DepartmentofMathematicalSciences TsinghuaUniversityBeijing100084P.R.China DepartmentofMathematics WuhanUniversityWuhan430072P.R.China

Two characterizations for a local diffeomorphism of R^n to be global one aregiven in terms of associated Wazewski equations. The two characterizations could be useful for theinvestigation of the Jacobian conjecture.

关键词： completeness diffeomorphism Jacobian conjecture properness

ON THE SOLUTION OF NONLINEARTWOPOINT BOUNDARY VALUE PROBLEM──u" + g(t.u) = f(t),u(0) = u(2π) = 0

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Applied Mathematics and Mechanics(English Edition) 1998年第9期19卷 889-894页

作者：黄文华曹菊生沈祖和 Wuxi Univ of Light Industry Wuxi China

In this paper, a non-variational version of a max-min principle is proposed, andan existence and uniqueness result is obtained for the nonlinear two-point boundaryvalue problenl u' + g(t.u) = f(t),u(0) = u(2π) = 0

关键词： Hilbert space diffeomorphism nonlinear two-point boundaryvalue problem unique solution

Vacuum Black Hole Mass Formula Is a Vanishing Noether Charge

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Communications in Theoretical Physics 2002年第9期38卷 309-312页

作者： WU Xiao-Ning GUO Han-Ying HUANG Chao-Guang WU Ke Institute of Theoretical Physics the Chinese Academy of Sciences P.O. Box 2735 Beijing 100080 China Institute of Theoretical Physics the Chinese Academy of Sciences P.O. Box 2735 Beijing 100080 China Institute of High Energy Physics the Chinese Academy of Sciences P.O. Box 9184 Beijing 100039 China Institute of Theoretical Physics the Chinese Academy of Sciences P.O. Box 2735 Beijing 100080 China Institute of Theoretical Physics the Chinese Academy of Sciences P.O. Box 2735 Beijing 100080 China Departrnent of Mathematics Capital Normal University Beijing 100037 China

The Noether current and its variation relation with respect to diffeomorphism invariance of gravitationaltheories have been derived from the horizontal variation and vertical-horizontal bi-variation of the Lagrangian, respec-tively. For Einstein's GR in the stationary, axisymmetric black holes, the mass formula in vacuum can be derived fromthis Noether current although it definitely vanishes. This indicates that the mass formula of black holes is a vanishingNoether charge in this case. The first law of black hole thermodynamics can also be derived from the variation relationof this vanishing Noether current.

关键词： black hole, mass formula, Noether charge, diffeomorphism invariance

Rigidity result on conjugacies of families of diffeomorphisms

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Science China Mathematics 1997年第10期40卷 1036-1044页

作者：李伟固章梅荣 Department of Mathematics Peking UniversityBeijing 100871China Department of Applied Mathematics Tsinghua UniversityBeijing 100084China

Embedding flows are used to obtain a rigidity result on strongly topological conjugacy of families of diffeomorphisms,i.***.families of C4(2≤r≤∞) diffeomorphisms,the strongly topologically conjugating homeomor-phisms... 详细信息

关键词： diffeomorphism topological conjugacy.