Fixed point sets of parabolic isometries of CAT(0)-spaces

Rotational and Parabolic Surfaces in PSL2(R,tau) and Applications

We study surfaces of constant mean curvature which are invariant by oneparameter group of either rotational isometries or parabolic isometries, immersed into the homogeneous manifold PSL2(R,tau). Also, we give some applications.

Parabolic Weingarten surfaces in hyperbolic space

A surface in hyperbolic space $\h^3$ invariant by a group of parabolic isometries is called a parabolic surface. In this paper we investigate parabolic surfaces of $\h^3$ that satisfy a linear Weingarten relation of the form $aκ_1+bκ_2=c$ or $aH+bK=c$, where $a,b,c\in \r$ and, as usual, $κ_i$ are the principal curvatures, $H$ is the mean curvature and $K$ is de Gaussian curvature. We classify all parabolic linear Weingarten surfaces in hyperbolic space.

horospherical products的准等距几何刚性

Geometric rigidity of quasi-isometries in horospherical products

本文证明了超几何空间的horospherical products的准等距在几何上是刚性的，即它们接近于乘积映射。研究了可解李群R ___ (N 1 x N 2)的准等距不变量和分类。

Quasicircle boundaries and exotic almost-isometries

We consider properly discontinuous, isometric, convex cocompact actions of surface groups on a CAT(-1) space. We show that the limit set of such an action, equipped with the canonical visual metric, is a (weak) quasicircle in the sense of Falconer and Marsh. It follows that the visual metrics on such limit sets are classified, up to bi-Lipschitz equivalence, by their Hausdorff dimension. This result applies in particular to boundaries at infinity of the universal cover of a locally CAT(-1) surface. We show that any two periodic CAT(-1) metrics on $\mathbb H^2$ can be scaled so as to be almost-…

Geometry and Dynamics of Discrete Isometry Groups of Higher Rank Symmetric Spaces

For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For symmetric spaces of higher rank, which contain isometrically embedded Euclidean planes, the situation becomes far more complicated. This paper is devoted to the study of the geometric limit set of ``nonelementary'' discrete isometry groups of higher rank symmetric spaces. We obtain…

Twisted hyperkähler symmetries and hyperholomorphic line bundles

In this paper we propose and investigate in full generality new notions of (continuous, non-isometric) symmetry on hyperkähler spaces. These can be grouped into two categories, corresponding to the two basic types of continuous hyperkähler isometries which they deform: tri-Hamiltonian isometries, on one hand, and rotational isometries, on the other. The first category of deformations gives rise to Killing spinors and generate what are known as hidden hyperkähler symmetries. The second category gives rise to hyperholomorphic line bundles over the hyperkähler manifolds on which they are defined …