Length Spectrums of Riemann Surfaces and the Teichmuller Metric

Suppose that T (S₀) is the Teichmüller space of a compactRiemann surface S₀ of genus g > 1. Let d_T(·, ·)be the Teichmüller metric of T(S₀) and let d_S(·,·) be a metric of T(S₀) defined by the length spectrumsof Riemann surfaces. The author showed in a previous paper thatd_T and d_S are topologically equivalent and , where C(₁) is a constant depending on ₁. In thispaper, it is shown that d_T and d_S are not metrically equivalent;that is, there is no constant C > 0 such that for all ₁ and ₂ in T(S₀). 2000 Mathematics SubjectClassification 32G15, 30C62, 30C75.     

Regularity of Sets with Quasiminimal Boundary Surfaces in Metric Spaces

This paper studies regularity of perimeter quasiminimizing sets in metric measure spaces with a doubling measure and a Poincaré inequality. The main result shows that the measure-theoretic boundary of a quasiminimizing set coincides with the topological boundary. We also show that such a set has finite Minkowski content and apply the regularity theory to study rectifiability issues related to quasiminimal sets in the strong A _∞-weighted Euclidean case.     

The Ricci Flow with Metric Torsion on Closed Surfaces

The famous Uniformization Theorem states that on closed Riemannian surfaces there always exists a metric of constant curvature for the Levi-Civita connection. In this article, we prove that an analogue of the uniformization theorem also holds for connections with metric torsion in the case of non-positive Euler characteristic. Our main tool is an adapted form of the Ricci flow.     

A Metric Characteristic of Minimal Surfaces on Arbitrary Carnot Groups

Mathematical Notes -     

Laguerre Geometry of Surfaces in R^3

Let f ： M → R3 be an oriented surface with non-degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L（f） = f（H2 - K）/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R^3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R^3. And we give a classification theorem of surfaces in R^3 with vanishing Laguerre form.     

度量投影的收敛性（英文）

讨论了赋范空间中度量投影的收敛性．得到了在局部紧集控制下，Chebyshev凸集序列的度量投影的收敛性与K－M收敛，Wijsman收敛和Kuratowski收敛都等价．本文的结论完善了M．Tsukada在[1]和[2]结果．     

A Class of Surfaces with Flat Blaschke Metric and Their Characterization

We study locally strongly convex surfaces with complete flat Blaschke metric. We show how we can characterize all known examples by a tensorial condition involving the covariant derivative of the shape operator and the gradient of the Pick invariant.Mathematics Subject Classifications (2000): 53A15.     

Bergman度量的完备性

本文证明了两个定理(1)设DcCn是一个完备的圆型域,若λ(D∪D)cD(0≤|λ|＜1),且对任意p∈D,有     

On the Norm of the Metric Projections

LetXbe a Banach space. GivenMa subspace ofXwe denote withP_Mthe metric projection ontoM. We defineπ(X) sup{P_M: Ma proximinal subspace ofX}. In this paper we give a bound forπ(X). In particular, whenX=L_p, we obtain the inequality P_M2^|2/p−1|, for every subspaceMofL_p. We also show thatπ(X)=π(X*).     

On Metric Boundedness Structures

Many years ago, S.-T. Hu gave necessary and sufficient conditions for a family of subsets of a metrizable space X to be the family of bounded sets for some admissible metric for the space. In this article, we show that in any noncompact metrizable space there are uncountably many distinct metric boundedness structures. Also, given an initial metric d for X, we look carefully at the problem of characterizing those boundedness structures determined by metrics uniformly equivalent to d. Applications to hyperspaces are given. Throughout, we rely on a dual approach to the study of metric boundedness.     

On Fuzzy Metric Space

In this paper we prove a common fixed point theorem for three mappings in fuzzy metric space and then extend this result to fuzzy 2 and 3-metric spaces. Our theorem is an extension of result of Fisher [12], to fuzzy metric spaces.AMS Subject Classification (1990): 47H10, 54H25     

On the Curvature of Metric Contact Pairs

We consider manifolds endowed with metric contact pairs for which the two characteristic foliations are orthogonal. We give some properties of the curvature tensor and in particular a formula for the Ricci curvature in the direction of the sum of the two Reeb vector fields. This shows that metrics associated to normal contact pairs cannot be flat. Therefore flat non-Kähler Vaisman manifolds do not exist. Furthermore we give a local classification of metric contact pair manifolds whose curvature vanishes on the vertical subbundle. As a corollary we have that flat associated metrics can only exist if the leaves of the characteristic foliations are at most three-dimensional.     

On the Metric Dimension of Infinite Graphs

A set of vertices S resolves a connected graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we undertake the metric dimension of infinite locally finite graphs, i.e., those infinite graphs such that all its vertices have finite degree. We give some necessary conditions for an infinite graph to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some general results about the metric dimension of the Cartesian product of finite and infinite graphs, and obtain the metric dimension of the Cartesian product of several families of graphs.     

On the Weak-Open Images of Metric Spaces

In this paper, we give characterizations of certain weak-open images of metric spaces.     

On Uniform Lipschitz-Connectedness in Metric Spaces

We show that the category of uniformly Lipschitz-connected metric spaces and Lipschitz maps is coreflective in the category of Lipschitz-connected metric spaces and Lipschitz maps.     

On Connection Between the Nonholonomic Metric on the Heisenberg Group and the Grushin Metric

The purpose of this paper is to compute geodesics on the Grushin plane and examine an assertion on connection between spheres of the Grushin plane and spheres of the Heisenberg group. The assertion turns out to require correction that the spheres of the Heisenberg group are directly obtained by rotation of the Grushin spheres. We find a modified Grushin metric for which the last assertion holds. Also, we prove several theorems about connections between the Grushin plane and Heisenberg group.     

On the Convergence of the Variable Metric Algorithm

The variable metric algorithm is a frequently used method forcalculating the least value of a function of several variables.However it has been proved only that the method is successfulif the objective function is quadratic, although in practiceit treats many types of objective functions successfully. Thispaper extends the theory, for it proves that successful convergenceis obtained provided that the objective function has a strictlypositive definite second derivative matrix for all values ofits variables. Moreover it is shown that the rate of convergenceis super-linear.