Ricci flow of conformally compact metrics

In this paper we prove that given a smoothly conformally compact asymptotically hyperbolic metric there is a short-time solution to the Ricci flow that remains smoothly conformally compact and asymptotically hyperbolic. We adapt recent results of Schnürer, Schulze and Simon to prove a stability result for conformally compact Einstein metrics sufficiently close to the hyperbolic metric.     

Extremal polygonal quasiconformal mappings and biLipschitz mappings

We show that the extremal polygonal quasiconformal mappings are biLipschitz with respect to the hyperbolic metric in the unit disk.     

A new weighted metric: the relative metric II

In the first part of this investigation we generalized a weighted distance function of R.-C. Li's and found necessary and sufficient conditions for it being a metric. In this paper some properties of this so-called M-relative metric are established. Specifically, isometries and quasiconvexity results are derived. We also illustrate connections between our approach and generalizations of the hyperbolic metric.     

Comparison theorems for hyperbolic type metrics

The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.     

Metrics of Hyperbolic Type on Bounded Fractals

On the bounded Sierpinski gasket F we use the set of essential fixed points V ₀ as a boundary and consider the fractal Brownian motion on F killed in V ₀. The corresponding Dirichlet–Laplacian is described in terms of a kind of hyperbolic distance, a metric which explodes near the boundary. We consider Harnack inequalities, Green’s function estimates and (random) products of matrices defining the local energy of harmonic functions. Supported by the DFG research group ‘Spektrale Analysis, asymptotische Verteilungen und stochastische Dynamik.’     

John圆与双曲度量

利用双曲度量讨论了John圆的几何性质,借助于Gehring-Hayman不等式建立了John圆的一个充要条件.     

共形平坦的Randers度量

本文研究共形平坦的Randers 度量的性质. 证明了具有数量旗曲率的共形平坦的Randers 度量都是局部射影平坦的, 并且给出了这类度量的分类结果. 本文还证明了不存在非平凡的共形平坦且具有近迷向S 曲率的Randers 度量.     

Rigidity Theorems for Einstein–Thorpe Metrics Dedicated to my parents

We will prove that every Einstein–Thorpe metric on T ⁸ must be flat and that on compact oriented hyperbolic manifolds of dimension 8, every Einstein–Thorpe metric is a hyperbolic metric up to rescalings and diffeomorphisms.     

Revisiting the foundations of Barbilian?s metrization procedure

In the present work we prove that one of Barbilian?s theorems from 1960 regarding the metrization procedure in the plane admits a natural extension depending on a bilinear form and the relative position of two Apollonian hyperspheres. This result allows us to pursue two fundamental ideas. First, that all the distances with constant curvature can be described by Barbilian?s metrization principle. Secondly, that all the Riemannian metric corresponding to these distances can be obtained with the same unique procedure derived from the main theorem in the text (Theorem 2.5). We show how the hyperbolic metric of the disk, the hyperbolic metric on the exterior of the disk and the hyperbolic metric on the half-plane can be obtained in the same way using Theorem 2.5, which appears here for the first time and is an extension of a Barbilian classical result (Barbilian, 1960 [7]). Furthermore, we obtain metrics corresponding to quadratic forms with signature that includes minus. By considering the norms provided by either Lorentz or Minkowski (pseudo-)inner product as influence functions, two oscillant distances can be generated in some subsets of Lorentz or Minkowski plane. The extension of 1960 Barbilian?s theorem mentioned above allow us to obtain the metrics attached to these two Barbilian distances on corresponding subsets of Lorentz and Minkowski 2-dimensional spaces. The geometric study concludes that these metrics are generalized Lagrange metrics. A result concerning the distance induced by a Riemannian metric as a local Barbilian distance is also proved.     

Bilipschitz Embeddings of Metric Spaces into Space Forms

The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.     

单位球上全纯函数的高阶Schwarz-Pick估计*

作者给出了单位球B_n ? Cⁿ到凸区域?? C上全纯函数的高阶Schwarz-Pick估计. 通过引入双曲度量,得到了单位圆盘D到凸区域?上全纯函数的系数估计. 应用该系数估计结果,得到单位球B_n到?内的全纯函数的高阶Schwarz-Pick估计.特别地, 当?是单位圆盘或右半平面时, 得到的结果分别与熟知的结果是一致的.     

Integrably asymptotic affine homeomorphisms of the circle and Teichmüller spaces

A quasisymmetric homeomorphism of the unit circle S1 is called integrably asymptotic affine if it admits a quasiconformal extension into the unit disk so that its complex dilatation is square integrable in the Poincar metric on the unit disk. Let QS*(S1) be the space of such maps. Here we give some characterizations and properties of maps in QS(S1). We also show that QS*(S1)/M(O)b(S1) is the completion of Diff(S1)/M(O)b(S1) in the Weil-Petersson metric.     

The Geodesics of Metric Connections with Vectorial Torsion

The present note deals with the dynamics of metric connections with vectorial torsion, as already described by E. Cartan in 1925. We show that the geodesics of metric connections with vectorial torsion defined by gradient vector fields coincide with the Levi-Civita geodesics of a conformally equivalent metric. By pullback, this yields a systematic way of constructing invariants of motion for such connections from isometries of the conformally equivalent metric, and we explain in as much this result generalizes the Mercator projection which maps sphere loxodromes to straight lines in the plane. An example shows that Beltrami's theorem fails for this class of connections. We then study the system of differential equations describing geodesics in the plane for vector fields which are not gradients, and show among others that the Hopf–Rinow theorem does also not hold in general.     

非欧空间中双基本图形的度量方程及其应用

本文提出了n维球面型空间和双曲空间中双基本图形的概念,建立了球面型空间与双曲空间中双基图形的度量方程,并给出度量方程的一些应用.     

Prescribing Curvature Problems on the Bakry-Emery Ricci Tensor of a Compact Manifold with Boundary

t The authors consider the problem of conformally deforming a metric such that the k-curvature defined by an elementary symmetric function of the eigenvalues of the Bakry-Emery Ricci tensor on a compact manifold with boundary to a prescribed function. A consequence of our main result is that there exists a complete metric such that the Monge-Amp~re type equation with respect to its Bakry-Emery Ricci tensor is solvable, provided that the initial Bakry-Emery Ricci tensor belongs to a negative convex cone.     

A conformally invariant metric on Riemann surfaces associated with integrable holomorphic quadratic differentials

In this paper, we define a conformally invariant (pseudo-)metric on all Riemann surfaces in terms of integrable holomorphic quadratic differentials and analyze it. This metric is closely related to an extremal problem on the surface. As a result, we have a kind of reproducing formula for integrable quadratic differentials. Furthermore, we establish a new characterization of uniform thickness of hyperbolic Riemann surfaces in terms of invariant metrics.     

Gromov hyperbolicity of the and metrics

In this note it is shown that the metric is always Gromov hyperbolic, but that the metric is Gromov hyperbolic if and only if has exactly one boundary point. As a corollary we get a new proof for the fact that the quasihyperbolic metric is Gromov hyperbolic in uniform domains.

     

Manhattan orbifolds

We investigate a class of metrics for 2-manifolds in which, except for a discrete set of singular points, the metric is locally isometric to an L₁ (or equivalently L_∞) metric, and show that with certain additional conditions such metrics are injective. We use this construction to find the tight span of squaregraphs and related graphs, and we find an injective metric that approximates the distances in the hyperbolic plane analogously to the way the rectilinear metrics approximate the Euclidean distance.     

The relationship between the Einstein and Yang-Mills equations

In this paper we prove that special requirements to Yang-Mills equations on a 4-dimensional conformally connected manifold allow one to reduce them to a system of Einstein equations and additional ones that bind components of the energy-impulse tensor. We propose an algorithm that gives conditions for the embedding of the metric of the gravitational field into a special (uncharged) Yang-Mills conformally connected manifold. As an application of the algorithm, we prove that the metric of any Einstein space and the Robertson-Walker metric are embeddable into the specified manifold.     

Isometries of the Half-Apollonian Metric

The half-Apollonian metric is a generalization of the hyperbolic metric, similar to the Apollonian metric. It can be defined in arbitrary domains in the euclidean space and has the advantages of being easy to calculate and estimate. We show that the half-Apollonian metric has many geodesics and use this fact to show that in most domains all the isometries of the metric are similarity mappings.