THE SOLUTION OF _b－EQUATION OF （P，Q）－FORMS AND IT＇S L~P＆HOLDER ESTIMATES ON A STEIN MANIFOLD

ＴＨＥＳＯＬＵＴＩＯＮＯＦ_ｂ－ＥＱＵＡＴＩＯＮＯＦ（Ｐ，Ｑ）－ＦＯＲＭＳＡＮＤＩＴ＇ＳＬ￣Ｐ＆ＨＯＬＤＥＲＥＳＴＩＭＡＴＥＳＯＮＡＳＴＥＩＮＭＡＮＩＦＯＬＤ￥ＷｕＸｉａｏｑｉｎ（ＤｅｐｔｏｆＭａｔｈ，ＪｉｍｅｉＴｅａｃｈｅｒｓＣｏｌｌｅｇｅＸｉａ?..     

ON THE EXISTENCE AND UNIQUENESS THEOREMS OF SOLUTIONS FOR A CLASS OF THE SYSTEMS OF MIXED MONOTONE OPERATOR EQUATIONS WITH APPLICATION

ＯＮＴＨＥＥＸＩＳＴＥＮＣＥＡＮＤＵＮＩＱＵＥＮＥＳＳＴＨＥＯＲＥＭＳＯＦＳＯＬＵＴＩＯＮＳＦＯＲＡＣＬＡＳＳＯＦＴＨＥＳＹＳＴＥＭＳＯＦＭＩＸＥＤＭＯＮＯＴＯＮＥＯＰＥＲＡＴＯＲＥＱＵＡＴＩＯＮＳＷＩＴＨＡＰＰＬＩＣＡＴＩＯＮＳＨＥＮＰＥＩＬＯＮＧ...     

SINGULAR INTEGRAL OPERATORS AND SINGULAR QUADRATURE OPERATORS ASSOCIATED WITH SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND AND THEIR APPLICATIONS

ＳＩＮＧＵＬＡＲＩＮＴＥＧＲＡＬＯＰＥＲＡＴＯＲＳＡＮＤＳＩＮＧＵＬＡＲＱＵＡＤＲＡＴＵＲＥＯＰＥＲＡＴＯＲＳＡＳＳＯＣＩＡＴＥＤＷＩＴＨＳＩＮＧＵＬＡＲＩＮＴＥＧＲＡＬＥＱＵＡＴＩＯＮＳＯＦＴＨＥＦＩＲＳＴＫＩＮＤＡＮＤＴＨＥＩＲＡＰＰＬＩＣＡＴＩ...     

THE NON－EXISTENCE OF PERIODIC SOLUTIONS OF A CERTAIN CLASS OF N－TH ORDER DIFFERENTIAL EQUATIONS

ＴＨＥ　ＮＯＮ－ＥＸＩＳＴＥＮＣＥ　ＯＦ　ＰＥＲＩＯＤＩＣ　ＳＯＬＵＴＩＯＮＳ　ＯＦ　Ａ　ＣＥＲＴＡＩＮ　ＣＬＡＳＳ　ＯＦ　Ｎ－ＴＨ　ＯＲＤＥＲ　ＤＩＦＦＥＲＥＮＴＩＡＬ　ＥＱＵＡＴＩＯＮＳＴＨＥＮＯＮ－ＥＸＩＳＴＥＮＣＥＯＦＰＥＲＩＯＤＩＣＳＯＬＵ...     

ELLIPTIC H~2－VOLTERRA PROJECTION AND THE H~1－GALERKIN METHODS FOR THE INTEGRO－DIFFERENTIAL EQUATIONS OF EVOLUTION

ＥＬＬＩＰＴＩＣＨ~２－ＶＯＬＴＥＲＲＡＰＲＯＪＥＣＴＩＯＮＡＮＤＴＨＥＨ~１－ＧＡＬＥＲＫＩＮＭＥＴＨＯＤＳＦＯＲＴＨＥＩＮＴＥＧＲＯ－ＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＯＦＥＶＯＬＵＴＩＯＮ￥ＳＵＮＰＥＮＧＴＡＯＡｂｓｔｒａｃｔ：Ｔｈｉｓｐ...     

SHAPE PRESERVING PROPERTIES AND CONVERGENCE OF UNIVARIATE MULTIQUADRIC QUASI－INTERPOLATION

ＳＨＡＰＥＰＲＥＳＥＲＶＩＮＧＰＲＯＰＥＲＴＩＥＳＡＮＤＣＯＮＶＥＲＧＥＮＣＥＯＦＵＮＩＶＡＲＩＡＴＥＭＵＬＴＩＱＵＡＤＲＩＣＱＵＡＳＩ－ＩＮＴＥＲＰＯＬＡＴＩＯＮＷｕＺＯＮＧＭＩＮ（吴宗敏）（ＤｅｐｅｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，Ｆｕｄａ...     

GAMMA－MINIMAX ESTIMATORS FOR THE MEAN OF A MULTIVARIATE NORMAL DISTRIBUTION WITH PARTIALLY UNKNOWN COVARIANCE MATRIX

ＧＡＭＭＡ－ＭＩＮＩＭＡＸＥＳＴＩＭＡＴＯＲＳＦＯＲＴＨＥＭＥＡＮＯＦＡＭＵＬＴＩＶＡＲＩＡＴＥＮＯＲＭＡＬＤＩＳＴＲＩＢＵＴＩＯＮＷＩＴＨＰＡＲＴＩＡＬＬＹＵＮＫＮＯＷＮＣＯＶＡＲＩＡＮＣＥＭＡＴＲＩＸＣＨＥＮＬＡＮＸＩＮＧ（陈兰祥）（Ｄｅｐａｒｔ...     

一类特殊微分代数方程组的性质及应用

一类特殊微分代数方程组的性质及应用顾金生，胡显承（清华大学应用数学系）ＴＨＥＰＲＯＰＥＲＴＩＥＳＯＦＡＫＩＮＤＯＦＤＩＦＦＥＲＥＮＴＩＡＬ／ＡＬＧＥＢＲＡＩＣＥＱＵＡＴＩＯＮＳＡＮＤＴＨＥＩＲＡＰＰＬＩＣＡＴＩＯＮＳ￥ＧｕＪｉｎ－ｓｈｅｎｇ；ＨｕＸｉ...     

ADIABATIC INVARIANTS OF SLOWLY VARYING THREE－DIMENSIONAL SYSTEMS AND EXISTENCE OF INVARIANT TORI OF LOTKA－VOLTERRA EQUATION

ＡＤＩＡＢＡＴＩＣＩＮＶＡＲＩＡＮＴＳＯＦＳＬＯＷＬＹＶＡＲＹＩＮＧＴＨＲＥＥ－ＤＩＭＥＮＳＩＯＮＡＬＳＹＳＴＥＭＳＡＮＤＥＸＩＳＴＥＮＣＥＯＦＩＮＶＡＲＩＡＮＴＴＯＲＩＯＦＬＯＴＫＡ－ＶＯＬＴＥＲＲＡＥＱＵＡＴＩＯＮＬＩＪＩＢＩＮＺＨＡＯＸＩＡＯＨ...     

EXISTENCE，UNIQUENESS AND PROPERTIES OF THE SOLUTIONS OF A DEGENERATE PARABOLIC EQUATION WITH DIFFUSION－ADVECTION－ABSORPTION

ＥＸＩＳＴＥＮＣＥ，ＵＮＩＱＵＥＮＥＳＳＡＮＤＰＲＯＰＥＲＴＩＥＳＯＦＴＨＥＳＯＬＵＴＩＯＮＳＯＦＡＤＥＧＥＮＥＲＡＴＥＰＡＲＡＢＯＬＩＣＥＱＵＡＴＩＯＮＷＩＴＨＤＩＦＦＵＳＩＯＮ－ＡＤＶＥＣＴＩＯＮ－ＡＢＳＯＲＰＴＩＯＮ￥ＳＯＮＧＢＩＮＨＥＮＧ（宋...     

Higher eta-invariants

We define the higher eta-invariant of a Dirac-type operator on a nonsimply-connected closed manifold. We discuss its variational properties and how it would fit into a higher index theorem for compact manifolds with boundary. We give applications to questions of positive scalar curvature for manifolds with boundary, and to a Novikov conjecture for manifolds with boundary.Partially supported by the Humboldt Foundation and NSF grant DMS-9101920.     

The Linearized Calderón Problem on Complex Manifolds

In this note we show that on any compact subdomain of a Kähler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calderón problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of Kähler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot be treated by standard methods for the Calderón problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends earlier results from the case of Riemann surfaces to higher dimensional complex manifolds.     

A Diffeomorphism Classification of Generalized Witten Manifolds

We give a classification of a specific family of seven-dimensional manifolds, the generalized Witten manifolds up to homeomorphism and diffeomorphism. Using an approach suggested by J. Shaneson, we develop a modified surgery theory which fully classifies these manifolds. In contrast to previous approaches, this surgery theory is designed to be more readily applicable to higher dimensions. The family contains examples of manifolds which admit Einstein metrics and are homeomorphic but not diffeomorphic. These particular manifolds occur naturally in differential geometry and are of great interest to both differential geometers and physicists.     

THE DIFFERENTIAL INTEGRAL EQUATIONS ON SMOOTH CLOSED ORIENTABLE MANIFOLDS

1 IntroductionSillce tl1e limit value fOrlnula, viz. tl1e Plemelj fOrn1ula, of the Cauthe type integraJ withBochner-Martinelli kernel was proved in 1957[1], it has beell successfully used to the study Ofsingular i1ltegral equatious, solvi11g the 0b--equation, holomorphic extension, 0--closed exten-sion and C-R 111al1ifolds[2-51. Evideutly, the researcl1 of higher order singular integrals withBochuer-Martinelli kerllel itself also l1as important significallce. In 1952, J. Hadanmrd firstde…     

Gauss-Bonnet-Chern mass and Alexandrov-Fenchel inequality

This is a survey about our recent works on the Gauss-Bonnet-Chern (GBC) mass for asymptotically flat and asymptotically hyperbolic manifolds. We first introduce the GBC mass, a higher order mass, for asymptotically flat and for asymptotically hyperbolic manifolds, respectively, by using a higher order scalar curvature. Then we prove its positivity and the Penrose inequality for graphical manifolds. One of the crucial steps in the proof of the Penrose inequality is the use of an Alexandrov-Fenchel inequality, which is a classical inequality in the Euclidean space. In the hyperbolic space, we have established this new Alexandrov-Fenchel inequality. We also have a similar work for asymptotically locally hyperbolic manifolds. At the end, we discuss the relation between the GBC mass and Chern’s magic form.     

Asymptotic Dirichlet Problem for Harmonic Maps with Bounded Image

Existence and uniqueness is proved for asymptotic Dirichlet problems on Hadamard manifolds. This includes manifolds of bounded negative curvature and symmetric spaces of higher rank.     

Weak and strong fillability of higher dimensional contact manifolds

For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five), while also being obstructed by all known manifestations of “overtwistedness”. We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.     

On Smooth Maps with Finitely Many Critical Points

The minimum number of critical points of a small codimensionsmooth map between two manifolds is computed. Some partial resultsfor the case of higher codimension when the manifolds are spheresare also given.     

On the cusped fan in a planar portrait of a manifold

Stable maps into the plane are good tools to obtain “views” of higher dimensional manifolds. We introduce the planar portraits to define the “view” properly. To start studying their relation to manifolds, we restrict our attention to their basic piece called the cusped fan. Fibreing structures over the cusped fan are studied and given a geometric characterisation. As by-products, we supply various stable maps and planar portraits of closed manifolds. In particular, two infiniteness properties of planar portraits are shown by using these examples.     

Counting commensurability classes of hyperbolic manifolds

Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v ^v such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Moreover, by restricting attention to non-compact manifolds, our result implies the same growth type for the number of quasi-isometry classes of lattices in SO(n, 1). Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms.