On the metric structure of non-Kähler complex surfaces

We give a characterization of a locally conformally K?hler (l.c.K.) metric with parallel Lee form on a compact complex surface. Using the Kodaira classification of surfaces, we classify the compact complex surfaces admitting such structures. This gives a classification of Sasakian structures on compact three-manifolds. A weak version of the above mentioned characterization leads to an explicit construction of l.c.K. metrics on all Hopf surfaces. We characterize the locally homogeneous l.c.K. metrics on geometric complex surfaces, and we prove that some Inoue surfaces do not admit any l.c.K. metric. Received: 23 July 1998 / Revised: 2 June 1999     

集值动力系统中的混合性和混沌

设(X,d)是紧致度量空间.设(K,H)是X中所有非空紧子集所组成的空间,并赋予由d导出的Hausdorff度量H.主要探讨了拓扑动力系统(X,G)的混合性、混沌和集值动力系统(K,G)的混合性、混沌之间的关系,其中G是拓扑群.     

The Bando-Calabi-Futaki character as an obstruction to semistability

For a K?hler class of a compact connected complex manifold, the associated Bando-Calabi-Futaki character is known as an obstruction to the existence of a K?hler metric in that class with constant scalar curvature. The purpose of this paper is to show that, for an integral K?hler class, the Bando-Calabi-Futaki character is an obstruction also to semistability in the geometric invariant theory. Received: 13 March 2001 / Published online: 23 May 2002     

Banach空间中关于有界集的同时远达问题的适定性

本文研究Banach空间中关于有界集的同时远达问题的适定性,在集合的Hausdorff距离下,证明了：对自反局部一致凸Banach空间中的闭有界集K,使所有关于K的同时远达问题是适定的紧凸子集A全体在紧凸子集全体中是Gδ型集．     

Ricci flow on Kähler-Einstein surfaces

In this paper, we prove that if M is a K?hler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature, and the curvature is positive somewhere, then the K?hler-Ricci flow converges to a K?hler-Einstein metric with constant bisectional curvature. In a subsequent paper [7], we prove the same result for general K?hler-Einstein manifolds in all dimension. This gives an affirmative answer to a long standing problem in K?hler Ricci flow: On a compact K?hler-Einstein manifold, does the K?hler-Ricci flow converge to a K?hler-Einstein metric if the initial metric has a positive bisectional curvature? Our main method is to find a set of new functionals which are essentially decreasing under the K?hler Ricci flow while they have uniform lower bounds. This property gives the crucial estimate we need to tackle this problem. Oblatum 8-IX-2000 & 30-VII-2001?Published online: 19 November 2001     

ITERATION OF FIXED POINTS ON HYPERSPACES

ＩＴＥＲＡＴＩＯＮＯＦＦＩＸＥＤＰＯＩＮＴＳＯＮＨＹＰＥＲＳＰＡＣＥＳＨＵＴＨＡＫＹＩＮ＊ＨＵＡＮＧＪＵＩＣＨＩ＊ＭａｎｕｓｃｒｉｐｔｒｅｃｅｉｖｅｄＯｃｔｏｂｅｒ２２，１９９６．＊Ｄｅｐａｒｔｍｅｎｔｏｆｍａｔｈｅｍａｔｉｃｓ，ＴａｍｋａｎｇＵｎｉ...     

一致覆盖和度量空间的紧映象

本文利用一致覆盖的概念,讨论了度量空间的序列覆盖紧映象的结构.主要结果有: (1)空间X是局部可分度量空间的序列覆盖紧映象当且仅当X具有由cosmic子空间构成的一致sn网; (2)空间X是局部可分度量空间的序列覆盖,商紧映象当且仅当X是度量空间的序列覆盖,商紧映象且是局部cosmic空间.     

Structural properties of compact groups with measure-theoretic applications

Every compact group is Baire isomorphic to a product of compact metric spaces; the isomorphism takes the Haar measure on the group to a direct product measure. This topological connection between compact groups and products of compact metric spaces provides a unified treatment for (Baire) measures on compact groups and for measures on topological products of metric spaces.     

Factorization theorems for cohomological dimension

The well-known factorization theorems for covering dimension dim and compact Hausdorff spaces are here established for the cohomological dimension dim using a new characterization of dim In particular, it is proved that every mapping f: X → Y from a compact Hausdorff space X with to a compact metric space Y admits a factorization f = hg, where g: X → Z, h: Z → Y and Z is a metric compactum with . These results are applied to the well-known open problem whether . It is shown that the problem has a positive answer for compact Hausdorff spaces X if and only if it has a positive answer for metric compacta X.     

Invariant Measures and Lower Ricci Curvature Bounds

Potential Analysis - Given a metric measure space $(X,d,\mathfrak {m})$ that satisfies the Riemannian Curvature Dimension condition, RCD?(K,N), and a compact subgroup of isometries G ≤...     

A simple proof of the bernoullicity of ergodic automorphisms on compact abelian groups

Recently it was proved by D. Lind, and G. Miles and K. Thomas that every ergodic automorphism of a compact metric abelian group is Bernoullian. They reduce the problem to the finite-dimensional compact connected abelian group (solenoidal group), and then they use difficult methods in proving the case. By using ideas of Y. Katznelson we can give a proof, which is much simpler than the other extant proofs, for the solenoidal case.     

度量空间的序列覆盖边界紧映象(英文)

本文主要讨论了度量空间的序列覆盖边界紧映象.用序列商、序列覆盖或1-序列覆盖的纤维边界紧或有限来刻画具有sn网或弱基的空间.主要结果如下:(1)度量空间上的序列覆盖边界紧映射是1-序列覆盖映射;(2)空间X是度量空间的序列商边界紧映象当且仅当X是snf-第一可数空间;(3)空间X是度量空间的序列覆盖边界紧S映象当且仅当X有点可数sn-网.     

Homogeneous spaces of curvature bounded below

We prove that every locally connected quotient G/H of a locally compact, connected, first countable topological group G by a compact subgroup H admits a G-invariant inner metric with curvature bounded below. Every locally compact homogeneous space of curvature bounded below is isometric to such a space. These metric spaces generalize the notion of Riemannian homogeneous space to infinite dimensional groups and quotients which are never (even infinite dimensional) manifolds. We study the geometry of these spaces, in particular of non-negatively curved homogeneous spaces. Dedicated to the memory of A. D. Alexandrov     

Isomorphisms of spaces of bounded continuous functions

If S is a locally compact metric space, countable at infinity, and the metric space T is not locally compact, then the spaces C(S) and C(T) are not isomorphic.     

Approximative compactness of the algebraic sum of sets

Let X be a group with an invariant metric, A and B nonempty subsets of X with B compact. It is proved that if A is an existence set [1] (approximatively compact [2]) then A + B and B + A are existence sets (approximatively compact). An example is given of a one-dimensional linear metric space (with an invariant metric) in which there exist an approximatively compact set A and an element v such that A + v is not an existence set.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 55–60, January, 1978.     

Weil-Petersson metric in the moduli space of compact polarized Kähler-Einstein manifolds of zero first Chern class

We study the problem of computing the curvature of the Weil-Petersson metric of the moduli space of general compact polarized Kähler-Einstein manifolds of zero first Chern class. We use canonical lifting of vector fields from the moduli space to the total deformation space to obtain a formula for the curvature of the Weil-Petersson metric. From this formula we obtain negative bisectional curvature for certain directions. This formula also reprove and explain the recent result of Schumacher that the holomorphic sectional curvature of the Weil-Petersson metric for K3-surfaces and symplectic manifolds are negative.     

Extension of functions preserving the modulus of continuity

The problem of the extension of a real-valued function from a subset of a metric space to the entire space is treated. An extension operator preserving the modulus of continuity of a function is proposed and its properties are studied. An application to the problem of the trace of a locally Lipschitz function on a compact subset of a metric space is given. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 236–245, February, 1997. Translated by N. K. Kulman     

The projective rank of a Hermitian symmetric space: A geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, the usual rank, and the 2-number (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Revised version: 6 August 2001 / Published online: 4 April 2002     

On certain generalizations of the Hopf bifurcation

New sufficient conditions for the existence of generalized Hopf bifurcations are given in the context of asymptotically compact dynamical or semidynamical systems on a metric space. These conditions weaken the hypotheses of previous contributions to the subject. Mathematics Subject Classification (2000) 34K18, 74H60     

Spaces of Densely Continuous Forms

The set C(X,Y) of continuous functions from a topological space X into a topological space Y is extended to the set D(X,Y) of densely continuous forms from X to Y, such form being a kind of multifunction from X to Y. The topologies of pointwise convergence, uniform convergence, and uniform convergence on compact sets are defined for D(X,Y), for locally compact spaces X and metric spaces Y having a metric satisfying the Heine–Borel property. Under these assumptions, D(X,Y) with the uniform topology is shown to be completely metrizable. In addition, if X is compact, D(X,Y) is completely metrizable under the topology of uniform convergence on compact sets. For this latter topology, an Ascoli theorem is established giving necessary and sufficient conditions for a subset of D(X,Y) to be compact.