A tale of two conformally invariant metrics

The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic metric in the disk. We prove that the Harnack distance is never greater than the hyperbolic distance and if the two distances agree for one pair of distinct points, then either the domain is simply connected or it is conformally equivalent to the punctured disk.     

The Hyperbolic Geometric Flow on Riemann Surfaces

In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors, motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on ?² in certain class of metrics, one can always choose suitable initial velocity symmetric tensor such that the solution exists for all time, and the scalar curvature corresponding to the solution metric g _ij keeps uniformly bounded for all time; moreover, if the initial velocity tensor is suitably “large", then the solution metric g _ij converges to the flat metric at an algebraic rate. If the initial velocity tensor does not satisfy the condition, then the solution blows up at a finite time, and the scalar curvature R(t, x) goes to positive infinity as (t, x) tends to the blowup points, and a flow with surgery has to be considered. The authors attempt to show that, comparing to Ricci flow, the hyperbolic geometric flow has the following advantage: the surgery technique may be replaced by choosing suitable initial velocity tensor. Some geometric properties of hyperbolic geometric flow on general open and closed Riemann surfaces are also discussed.     

Hyperbolic Geometry and the Hillam–Thron Theorem

Every open ball within has an associated hyperbolic metric and Möbius transformations act as hyperbolic isometries from one ball to another. The Hillam–Thron Theorem is concerned with images of balls under Möbius transformation, yet existing proofs of the theorem do not make use of hyperbolic geometry. We exploit hyperbolic geometry in proving a generalisation of the Hillam–Thron Theorem and examine the precise configurations of points and balls that arise in that theorem.This work was supported by Science Foundation Ireland grant 05/RFP/MAT0003     

Exact Controllability and Perturbation Analysis for Elastic Beams

The Rayleigh beam is a perturbation of the Bernoulli–Euler beam. We establish convergence of the solution of the Exact Controllability Problem for the Rayleigh beam to the corresponding solution of the Bernoulli–Euler beam. Convergence is related to a Singular Perturbation Problem. The main tool in solving this perturbation problem is a weak version of a lower bound for hyperbolic polynomials.     

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.     

WEIERSTRASS REPRESENTATION FORSURFACES WITH PRESCRIBED NORMALGAUSS MAP AND GAUSS CURVATURE IN H~3

The author obtains a Weierstrass representation for surfaces with prescribed normal Gauss map and Gauss curvature in H3. A differential equation about the hyperbolic Gauss map is also obtained, which characterizes the relation among the hyperbolic Gauss map, the normal Gauss map and Gauss curvature. The author discusses the harmonicity of the normal Gauss map and the hyperbolic Gauss map from surface with constant Gauss curvature in H3 to S2 with certain altered conformal metric. Finally, the author considers the surface whose normal Gauss map is conformal and derives a completely nonlinear differential equation of second order which graph must satisfy.     

Characterizations of hyperbolically convex regions

Let X be a simply connected and hyperbolic subregion of the complex plane C. A proper subregion Ω of X is called hyperbolically convex in X if for any two points A and B in Ω, the hyperbolic geodesic arc joining A and B in X is always contained in Ω. We establish a number of characterizations of hyperbolically convex regions Ω in X in terms of the relative hyperbolic density ρΩ(w) of the hyperbolic metric of Ω to X, that is the ratio of the hyperbolic metric of Ω to the hyperbolic metric of X. Introduction of hyperbolic differential operators on X makes calculations much simpler and gives analogous results to some known characterizations for euclidean or spherical convex regions. The notion of hyperbolic concavity relative to X for real-valued functions on Ω is also given to describe some sufficient conditions for hyperbolic convexity.     

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.     

Systolic volume of hyperbolic manifolds and connected sums of manifolds

The systolic volume of a closed n-manifold M is defined as the optimal constant σ(M) satisfying the inequality vol(M, g) ≥ σ(M) sys(M, g) ⁿ between the volume and the systole of every metric g on M. First, we show that the systolic volume of connected sums of closed oriented essential manifolds is unbounded. Then, we prove that the systolic volume of every sequence of closed hyperbolic (three-dimensional) manifolds is also unbounded. These results generalize systolic inequalities on surfaces in two different directions.      

Mixed initial boundary value problem for hyperbolic geometric flow on Riemann surfaces

The hyperbolic geometric flow equations is introduced recently by Kong and Liu motivated by Einstein equation and Hamilton Ricci flow. In this paper, we consider the mixed initial boundary value problem for hyperbolic geometric flow, and prove the global existence of classical solutions. The results show that, for any given initial metric on R² in certain class of metric, one can always choose suitable initial velocity symmetric tensor such that the solutions exist, and the scalar curvature corresponding to the solution metric gij keeps bounded. If the initial velocity tensor does not satisfy the certain conditions, the solutions will blow up at a finite time. Some special explicit solutions to the reduced equation are given.     

Adiabatic Limit for Hyperbolic Ginzburg–Landau Equations

We study the adiabatic limit in hyperbolic Ginzburg–Landau equations which are Euler–Lagrange equations for the Abelian Higgs model. Solutions of Ginzburg–Landau equations in this limit converge to geodesics on the moduli space of static solutions in the metric determined by the kinetic energy of the system. According to heuristic adiabatic principle, every solution of Ginzburg–Landau equations with sufficiently small kinetic energy can be obtained as a perturbation of some geodesic. A rigorous proof of this result was proposed recently by Palvelev.     

New perturbation bounds and condition numbers for the hyperbolic QR factorization

Using the modified matrix-vector equation approach, the technique of Lyapunov majorant function and the Banach fixed point theorem, we obtain some new rigorous perturbation bounds for R factor of the hyperbolic QR factorization under normwise perturbation. These bounds are always tighter than the one given in the literature. Moreover, the optimal first-order perturbation bounds and the normwise condition numbers for the hyperbolic QR factorization are also presented.     

Geodesic ideal triangulations exist virtually

It is shown that every non-compact hyperbolic manifold of finite volume has a finite cover admitting a geodesic ideal triangulation. Also, every hyperbolic manifold of finite volume with non-empty, totally geodesic boundary has a finite regular cover which has a geodesic partially truncated triangulation. The proofs use an extension of a result due to Long and Niblo concerning the separability of peripheral subgroups.

     

Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds

In this paper we prove a result on lower semicontinuity of pullback attractors for dynamical systems given by semilinear differential equations in a Banach space. The situation considered is such that the perturbed dynamical system is non-autonomous whereas the limiting dynamical system is autonomous and has an attractor given as union of unstable manifold of hyperbolic equilibrium points. Starting with a semilinear autonomous equation with a hyperbolic equilibrium solution and introducing a very small non-autonomous perturbation we prove the existence of a hyperbolic global solution for the perturbed equation near this equilibrium. Then we prove that the local unstable and stable manifolds associated to them are given as graphs (roughness of dichotomy plays a fundamental role here). Moreover, we prove the continuity of this local unstable and stable manifolds with respect to the perturbation. With that result we conclude the lower semicontinuity of pullback attractors.     

Hyperbolic metric and normal family

In this paper, we study the normality of the family of meromorphic functions from the viewpoint of hyperbolic metric. Then, a new sufficient and necessary condition is obtained, which can determine a given family of meromorphic functions is normal or not.     

关于双曲空间形式的一个注记

本文主要证明一个具有光滑边界的紧黎曼流形,如果有非平凡解,则就等度量同构与双曲空间形式 会的紧区域,这里Ｄ～２■是■的Ｈｅｓｓｉａｎ与ｇ是Ｍ上的黎曼度量． 还证明关于上述方程的边值问题,只有混合边值问题,而且当ｃ＜－１时有解．     

Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls

We show that the open unit ball of the space of operators from a finite-dimensional Hilbert space into a separable Hilbert space (we call it “operator ball”) has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In Appendix A we present results of Itai Shafrir about hyperbolic metrics on the operator ball.     

Hyperbolization of cusps with convex boundary

We prove that for every metric on the torus with curvature bounded from below by ?1 in the sense of Alexandrov there exists a hyperbolic cusp with convex boundary such that the induced metric on the boundary is the given metric. The proof is by polyhedral approximation. This was the last open case of a general theorem: every metric with curvature bounded from below on a compact surface is isometric to a convex surface in a 3-dimensional space form.     

Hyperbolic embeddedness and extension–convergence theorems of J-holomorphic curves

First, we give some characterization of hyperbolic embeddedness in the almost complex case. Next, we study the stability of hyperbolically embedded manifolds under a small perturbation of almost complex structures. Finally, we obtain generalizations and extensions of theorems of Kobayashi, Kiernan, Kwack and Noguchi for almost complex manifolds.     

Geometrically infinite surfaces with discrete length spectra

It is well-known that the length spectrum of a geometrically finite hyperbolic manifold is discrete. In this paper, we begin a study of the length spectrum for geometrically infinite hyperbolic surfaces. In this generality, it is possible that the spectrum is not discrete and the main focus of this work is to find necessary and sufficient conditions for a geometrically infinite surface to have a discrete spectrum. After deriving a number of properties of the length spectrum, we show that every topological surface of infinite type admits both an infinite dimensional family of quasiconformally distinct hyperbolic structures having a discrete length spectrum, and an infinite dimensional family of quasiconformally distinct structures with a nondiscrete spectrum. Moreover, there exists such an infinite dimensional subspace arbitrarily close to (in the Fenchel-Nielsen topology) any hyperbolic structure.