The Minimal Length of a Closed Geodesic Net on a Riemannian Manifold with a Nontrivial Second Homology Group

Let Mⁿ be a closed Riemannian manifold with a nontrivial second homology group. In this paper we prove that there exists a geodesic net on Mⁿ of length at most 3 diameter(Mⁿ). Moreover, this geodesic net is either a closed geodesic, consists of two geodesic loops emanating from the same point, or consists of three geodesic segments between the same endpoints. Geodesic nets can be viewed as the critical points of the length functional on the space of graphs immersed into a Riemannian manifold. One can also consider other natural functionals on the same space, in particular, the maximal length of an edge. We prove that either there exists a closed geodesic of length ≤ 2 diameter(Mⁿ), or there exists a critical point of this functional on the space of immersed θ-graphs such that the value of the functional does not exceed the diameter of Mⁿ. If n=2, then this critical θ-graph is not only immersed but embedded.Mathematics Subject Classifications (2000). 53C23, 49Q10     

Rigidity for critical metrics of the volume functional

Geodesic balls in a simply connected space forms , or are distinguished manifolds for comparison in bounded Riemannian geometry. In this paper we show that they have the maximum possible boundary volume among Miao–Tam critical metrics with connected boundary provided that the boundary of the manifold has a lower bound for the Ricci curvature. In the same spirit we also extend a rigidity theorem due to Boucher et al. 7 and Shen 18 to n‐dimensional static metrics with positive constant scalar curvature, which gives us a partial answer to the Cosmic no‐hair conjecture.     

The Distribution of Geodesic Excursions Out the End of a Hyperbolic Orbifold and Approximation with Respect to a Fuchsian Group

A generic geodesic on a finite area, noncompact surface with a hyperbolic orbifold structure travels densely about the surface, making excursions out the non-compact end and then returning to a compact piece of the surface. We show that for almost all geodesics on such a surface there is a limiting distribution for the sequence of depths of the excursions out a noncompact end. The distribution is independent of the surface and closely resembles the distribution arising from a sequence of continued fraction approximations in the metrical theory of diophantine approximation. This metrical theory can then be reproduced in the setting of approximation by parabolic fixed points in a Fuchsian group.     

可参数化曲面上特征值比较的数值计算原理

对于一个给定的完备可参数化曲面,我们给出了其上特征值比较时进行数值计算所遵循的基本原理.     

Banach空间中一类序压缩映射的不动点定理

在Banach空间中,利用迭代方法,研究了满足一定条件的序压缩算子的一些性质,获得了一类序压缩映射的不动点定理,证明了相应的结果,推广和改进了原有的结论,使其应用范围更加广泛.     

The matching theorems and coincidence theorems for generalized R-KKM mapping in topological spaces

In this paper we present some new matching theorems with open cover and closed cover by using the generalized R-KKM theorems [L. Deng, X. Xia, Generalized R-KKM theorem in topological space and their applications, J. Math. Anal. Appl. 285 (2003) 679-690] in the topological spaces with property (H). As applications, some coincidence theorems are established in topological spaces. Our results extend and generalize some known results.     

The semiclassical resolvent and the propagator for non-trapping scattering metrics

Consider a compact manifold with boundary M with a scattering metric g or, equivalently, an asymptotically conic manifold (M^○,g). (Euclidean Rn, with a compactly supported metric perturbation, is an example of such a space.) Let Δ be the positive Laplacian on (M,g), and V a smooth potential on M which decays to second order at infinity. In this paper we construct the kernel of the operator ⁻¹(h²Δ+V−²(λ₀±i0)), at a non-trapping energy λ₀>0, uniformly for h∈(0,h₀), h₀>0 small, within a class of Legendre distributions on manifolds with codimension three corners. Using this we construct the kernel of the propagator, e−it(Δ/2+V), t∈(0,t₀) as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.     

Boundary of Anosov dynamics and evolution equations for surfaces

We show that a compact surface of genus greater than one, without focal points and a finite number of bubbles (“good” shaped regions of positive curvature) is in the closure of Anosov metrics. Compact surfaces of nonpositive curvature and genus greater than one are in the closure of Anosov metrics, by Hamilton's work about the Ricci flow. We generalize this fact to the above surfaces without focal points admitting regions of positive curvature using a “magnetic” version of the Ricci flow, the so‐called Ricci Yang‐Mills flow.     

The asymptotic eigenfunctions of the operator ?D(x, y)? corresponding to Liouville metrics and waves on water captured by bottom irregularities

We construct a set of examples of bottom reliefs for which there exist captured waves corresponding to quasimodes of the wave operator ∇D(x, y)∇. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 414–422, September, 1998. The author thanks S. Yu. Dobrokhotov for stating the problem and numerous helpful discussions.     

On the curvature of the diffeomorphisms group

An expression for the sectional curvature ofSDIFF(M) (the group of diffeomorphism preserving Riemannian density on a closed manifoldM) is obtained. In the case of a locally Euclidean manifoldM, the negativeness of curvature that implies the instability of solutions of Euler equations of ideal incompressible fluids onM is established.     

Banach空间中线性流形上的度量投影的表达式

借助于正规对偶映射,建立了一般Banach空间中线性流形上的(集值)度量投影存在的 充要条件,同时给出了度量投影的表达式和点到线性流形上的距离公式.这些本质地推广和改进了 王玉文和于金凤在空间自反、严格凸和光滑强假定下的相应结果.     

Invariance of the Nayatani metrics for Kleinian manifolds

The Nayatani metric g _N is a Riemannian metric on a Kleinian manifold M which is compatible with the standard flat conformal structure. It is known that, for M corresponding to a geometrically finite Kleinian group, g _N has large symmetry: the isometry group of (M, g _N ) coincides with the conformal transformation group of M. In this paper, we prove that this holds for a larger class of M. In particular, this class contains such M that correspond to Kleinian groups of divergence type.      

The Einstein-Kahler metrics on Cartan-Hartogs domain of the first type

Let YI be the Cartan-Hartogs domain of the first type. We give the generating function of the Einstein-Kahler metrics on YI, the holomorphic sectional curvature of the invariant Einstein-Kahler metrics on YI. The comparison theorem of complete Einstein-Kahler metric and Kobayashi metric on YI is provided for some cases. For the non-homogeneous domain YI, when K =mn+1/m+n,m>1, the explicit forms of the complete Einstein-Kahler metrics are obtained.     

A mapping technique for numerical computations of bed evolutions

Free surface flow is one of the most difficult problems in engineering to be solved, since velocity and pressure fields depend on the free surface. On the other hand, the position of the free surface is unknown previously. Furthermore, the boundary condition on the free surface is expressed by a complicated equation. In an alluvial stream, where the boundaries of the domain are not fixed, addition of free surface at the bed will increase this difficulty. A domain mapping technique is developed in this paper to study the bed evolutions. The flow is considered 2D, choosing two coordinates in streamwise and upward directions. With a proper transformation, the hydrodynamics and sediment transport governing equations in irregular domain will be mapped into a simple rectangular one. The new domain can be discretize by finite elements. The transformed governing equations are solved to obtain desired variables in the mapped domain. With a proper transformation, there is no need of inverse mapping to obtain the free water surface profile and bedform evolution and migration in the actual domain. The model has been applied to streams with movable bed and the results show a good agreement with the experimental experiences.     

Extremal metrics for the first eigenvalue of the Laplacian in a conformal class

Let be a compact manifold. First, we give necessary and sufficient conditions for a Riemannian metric on to be extremal for with respect to conformal deformations of fixed volume. In particular, these conditions show that for any lattice of , the flat metric induced on from the standard metric of is extremal (in the previous sense). In the second part, we give, for any , an upper bound of on the conformal class of and exhibit a class of lattices for which the metric maximizes on its conformal class.

     

Extremal <Emphasis Type="Italic">G</Emphasis>-invariant eigenvalues of the Laplacian of <Emphasis Type="Italic">G</Emphasis>-invariant metrics

The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere S ² endowed with S ¹-invariant metrics, we consider the subsequence of the spectrum of a Riemannian manifold M which corresponds to metrics and functions invariant under the action of a compact Lie group G. If G has dimension at least 1, we show that the functional λ _k ^G admits no extremal metric under volume-preserving G-invariant deformations. If, moreover, M has dimension at least three, then the functional is unbounded when restricted to any conformal class of G-invariant metrics of fixed volume. As a special case of this, we can consider the standard O(n)-action on S ⁿ ; however, if we also require the metric to be induced by an embedding of S ⁿ in , we get an optimal upper bound on .