Chinese Annals of Mathematics, Series B

Items from 1 to 20 out of 872 results

article

η-Invariant and Flat Vector Bundles*

Xiaonan Ma, Weiping Zhang

Chinese Annals of Mathematics, Series B > 2006 > 27 > 1 > 67-72

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 real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah, Patodi...

article

Blow-up Mechanism of Classical Solutions to Quasilinear Hyperbolic Systems in the Critical Case*

Tatsien Li, Libin Wang

Chinese Annals of Mathematics, Series B > 2006 > 27 > 1 > 53-66

This paper deals with the blow-up phenomenon, particularly, the geometric blow-up mechanism, of classical solutions to the Cauchy problem for quasilinear hyperbolic systems in the critical case. We prove that it is still the envelope of the same family of characteristics which yields the blowup of classical solutions to the Cauchy problem in the critical case.

article

The Einstein-Scalar Field Constraints on Asymptotically Euclidean Manifolds*

Yvonne Choquet-Bruhat, James Isenberg, Daniel Pollack

Chinese Annals of Mathematics, Series B > 2006 > 27 > 1 > 31-52

By using the conformal method, solutions of the Einstein-scalar field 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...

article

SO(n)-Invariant Special Lagrangian Submanifolds of ℂⁿ⁺¹ with Fixed Loci*

Robert L. Bryant

Chinese Annals of Mathematics, Series B > 2006 > 27 > 1 > 95-112

Let SO(n) act in the standard way on ℂⁿ and extend this action in the usual way to ℂⁿ⁺¹ = ℂ ⊕ ℂⁿ. It is shown that a nonsingular special Lagrangian submanifold L ⊂ ℂⁿ⁺¹ that is invariant under this SO(n)-action intersects the fixed ℂ ⊂ ℂⁿ⁺¹ in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular...

article

Riemann-Finsler Geometry with Applications to Information Geometry

Zhongmin Shen

Chinese Annals of Mathematics, Series B > 2006 > 27 > 1 > 73-94

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.

article

On the Volume Formulas of Cones and Orthogonal Multi-cones in S ⁿ(1) and H ⁿ(—1)

Wu-Yi Hsiang

Chinese Annals of Mathematics, Series B > 2006 > 27 > 1 > 1-30

In the study of n-dimensional spherical or hyperbolic geometry, n≥ 3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with. In this paper, we use the method of infinitesimal symmetrization to provide a systematic way of obtaining volume formulas of cones and orthogonal multiple cones in S ⁿ(1) and H ⁿ(—1).

article

Associative Cones and Integrable System

Chuu-Lian Terng*, Shengli Kong, Erxiao Wang

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 153-168

We identify ℝ⁷ 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 ℝ⁷ 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...

article

Curvature, Diameter and Bounded Betti Numbers

Zhongmin Shen, Jyh-Yang Wu*

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 143-152

In this paper, we introduce the notion of bounded Betti numbers, and show that the bounded Betti numbers of a closed Riemannian n-manifold (M, g) with Ric (M) ≥ -(n - 1) and Diam (M) ≤ D are bounded by a number depending on D and n. We also show that there are only finitely many isometric isomorphism types of bounded cohomology groups $$ {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{H}^{*}...

article

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

Xu-Jia Wang

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 169-178

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 when the manifold is locally conformally flat or the Ricci curvature is positive.

article

A Geometric Problem and the Hopf Lemma. II

Yan Yan Li*, Louis Nirenberg

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 193-218

A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in ℝⁿ⁺¹, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane X _n+1 =constant in case M satisfies: for any two points (X′,X _n+1), $$ {\left( {{X}\ifmmode{'}\else$'$\fi,\ifmmode\expandafter\hat\else\expandafter\^\fi{X}_{{n...

article

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

Gang Tian*, Zhou Zhang

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 179-192

This note concerns the global existence and convergence of the solution for Kähler-Ricci flow equation when the canonical class, K _X, is numerically effective and big. We clarify some known results regarding this flow on projective manifolds of general type and also show some new observations and refined results.

article

Note on a Conjecture of Gopakumar-Vafa

Jun Li*, Baosen Wu

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 219-242

We rephrase the Gopakumar-Vafa conjecture on genus zero Gromov-Witten invariants of Calabi-Yau threefolds in terms of the virtual degree of the moduli of pure dimension one stable sheaves and investigate the conjecture for K3 fibred local Calabi-Yau threefolds.

article

Geometry of Ricci Solitons*

Huai-Dong Cao

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 121-142

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 solitons and the role they play in the singularity study of the Ricci flow.

article

Ricci Curvature and Fundamental Group*

Yuanlong Xin

Chinese Annals of Mathematics, Series B > 2006 > 27 > 2 > 113-120

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

article

Curvature Estimates for Irreducible Symmetric Spaces

Xusheng Liu

Chinese Annals of Mathematics, Series B > 2006 > 27 > 3 > 287-302

By making use of the classification of real simple Lie algebra, we get the maximum of the squared length of restricted roots case by case, and thus get the upper bounds of sectional curvature for irreducible Riemannian symmetric spaces of compact type. As an application, this paper verifies Sampson’s conjecture in most cases for irreducible Riemannian symmetric spaces of noncompact type.

article

Two New Families in the Stable Homotopy Groups of Sphere and Moore Spectrum*

Jinkun Lin

Chinese Annals of Mathematics, Series B > 2006 > 27 > 3 > 311-328

This paper proves the existence of an order p element in the stable homotopy group of sphere spectrum of degree p ⁿ q + p ^m q + q-4 and a nontrivial element in the stable homotopy group of Moore spectum of degree p ⁿ q + p ^m q + q-3 which are represented by h ₀(h _m b _n-1 - h _n b _m-1) and i _*(h ₀ h _n h _m) in the E ₂-terms of the Adams spectral sequence respectively, where p ≥ 7 is...

article

Geometric Optics for One-Dimensional Schrödinger-Poisson System

Zhifei Zhang, Qionglei Chen

Chinese Annals of Mathematics, Series B > 2006 > 27 > 3 > 353-362

This paper considers a family of Schrödinger-Poisson system in one dimension, whose initial data oscillates so that a caustic appears. By using the Lagrangian integrals, the authors obtain a uniform description of the solution outside the caustic, and near the caustic.

article

The Convergence of $$ \widetilde{\gamma }_{s} {\left( {b_{0} h_{n} - h_{1} b_{{n - 1}} } \right)} $$ *

Xiugui Liu, Xiangjun Wang

Chinese Annals of Mathematics, Series B > 2006 > 27 > 3 > 329-340

This paper computes the Thom map on γ₂ and proves that it is represented by 2b _2,0 h _1,2 in the ASS. The authors also compute the higher May differential of b _2,0, from which it is proved that $$ \widetilde{\gamma }_{s} {\left( {b_{0} h_{n} - h_{1} b_{{n - 1}} } \right)} $$ for 2 ≤ s < p - 1 are permanent cycles in the ASS.

article

On Completeness and Minimality of Random Exponential System in a Weighted Banach Space of Functions Continuous on the Real Line*

Zhiqiang Gao, Guantie Deng

Chinese Annals of Mathematics, Series B > 2006 > 27 > 3 > 303-310

In this paper, the completeness and minimality properties of some random exponential system in a weighted Banach space of complex functions continuous on the real line for convex nonnegative weight are studied. The results may be viewed as a probabilistic version of Malliavin's classical results.

article

A Hybrid of Theorems of Goldbach and Piatetski-Shapiro*

Xianmeng Meng, Mingqiang Wang

Chinese Annals of Mathematics, Series B > 2006 > 27 > 3 > 341-352

It is proved that for almost all sufficiently large even integers n, the prime variable equation n = p ₁ + p ₂, p ₁ ∈ P _γ is solvable, with 13=15 < γ ≤ 1, where P _γ = {p ∣ $$ p = {\left[ {m^{{\frac{1} {\gamma }}} } \right]} $$ ; for integer m and prime p} is the set of the Piatetski-Shapiro primes.

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GLOBAL EXISTENCE (11)
INVERSE PROBLEM (10)
CAUCHY PROBLEM (9)
STABILITY (9)
HOMOGENIZATION (8)
NAVIER-STOKES EQUATIONS (8)
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35L60 (5)
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35J60 (4)
53C21 (4)
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ASYMPTOTICS (4)
COMPRESSIBLE NAVIER-STOKES EQUATIONS (4)
CONTROLLABILITY (4)
CURVATURE ESTIMATES (4)
DATA ASSIMILATION (4)
EULER EQUATIONS (4)
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MAY SPECTRAL SEQUENCE (4)
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MIXED INITIAL-BOUNDARY VALUE PROBLEM (4)
NULL CONTROLLABILITY (4)
POINCARÉ INEQUALITY (4)
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UNIT BALL (4)
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VON NEUMANN ALGEBRAS (4)
WAVE EQUATION (4)
11R70 (3)
35B40 (3)
35L45 (3)
53C35 (3)
57M25 (3)
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BLOWUP (3)
BOUNDARY VALUE PROBLEMS (3)
CAPACITY (3)
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COHOMOLOGY (3)
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DEHN TWISTS (3)
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ELLIPTIC EQUATION (3)
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Chinese Annals of Mathematics, Series B

η-Invariant and Flat Vector Bundles*

Blow-up Mechanism of Classical Solutions to Quasilinear Hyperbolic Systems in the Critical Case*

The Einstein-Scalar Field Constraints on Asymptotically Euclidean Manifolds*

SO(n)-Invariant Special Lagrangian Submanifolds of ℂⁿ⁺¹ with Fixed Loci*

Riemann-Finsler Geometry with Applications to Information Geometry

On the Volume Formulas of Cones and Orthogonal Multi-cones in S ⁿ(1) and H ⁿ(—1)

Associative Cones and Integrable System

Curvature, Diameter and Bounded Betti Numbers

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

A Geometric Problem and the Hopf Lemma. II

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

Note on a Conjecture of Gopakumar-Vafa

Geometry of Ricci Solitons*

Ricci Curvature and Fundamental Group*

Curvature Estimates for Irreducible Symmetric Spaces

Two New Families in the Stable Homotopy Groups of Sphere and Moore Spectrum*

Geometric Optics for One-Dimensional Schrödinger-Poisson System

The Convergence of $$ \widetilde{\gamma }_{s} {\left( {b_{0} h_{n} - h_{1} b_{{n - 1}} } \right)} $$ *

On Completeness and Minimality of Random Exponential System in a Weighted Banach Space of Functions Continuous on the Real Line*

A Hybrid of Theorems of Goldbach and Piatetski-Shapiro*

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