Alexander Lytchak

Open map theorem for metric spaces

An open map theorem for metric spaces is proved and some applications are discussed. The result on the existence of gradient flows of semiconcave functions is generalized to a large class of spaces.

Metric characterizations of spherical and Euclidean buildings

A building is a simplicial complex with a covering by Coxeter complexes (called apartments) satisfying certain combinatorial conditions. A building whose apartments are spherical (respectively Euclidean) Coxeter complexes has a natural piecewise spherical (respectively Euclidean) metric with nice geometric properties. We show that spherical and Euclidean buildings are completely characterized by some simple, geometric properties. AMS Classication numbers Primary: 20E42 Secondary: 20F65

Differentiation in metric spaces

1.1. The Aim. This paper is devoted to the study of the first order geometry of metric spaces. Our study was mainly motivated by the observation that whereas the advanced features of the theories of Alexandrov spaces with upper and lower curvature bounds are quite different, the beginnings are almost identical, at least as far as only first order derivatives are concerned (for example tangent spaces and the first variation formula). One is naturally led to the question on which spaces the first order geometry can be established. As it turns out the same first order geometry exists in many other spaces that we call geometric. The class of geometric spaces contains all Holder continuous Riemannian manifolds, sufficiently convex and smooth Finsler manifolds ([LY]), a big class of subsets of Riemannian manifolds (for example sets of positive reach, see [Fed59] and [Lyta]), surfaces with an integral curvature bound ([Res93]) and extremal subsets of Alexandrov spaces with lower curvature bound ([PP94a]). The last case was discussed in [Pet94] and the proof of the first variation formula was a major step towards proving the deep gluing theorem ([Pet94]). Moreover the class of geometric spaces is stable under metric operations, even under such a difficult one as taking quotients. Finally the existence of the first order geometry is a good assumption for studying features of higher order, such as gradient flows of semi-concave functions ([PP94b] and [Lytc]). One of the main issues of this paper is the establishing of natural, easily verifiable axioms, that describe this first order geometry and their consequences.

At infinity of finite-dimensional CAT(0) spaces

We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space X has a non-empty intersection in the visual bordification

Area Minimizing Discs in Metric Spaces

{ overline{X} = X cup partial X}

Isometric actions on spheres with an orbifold quotient

. Using this fact, several results known for proper CAT(0) spaces may be extended to finite-dimensional spaces, including the existence of canonical fixed points at infinity for parabolic isometries, algebraic and geometric restrictions on amenable group actions, and geometric superrigidity for non-elementary actions of irreducible uniform lattices in products of locally compact groups.

Riemannian foliations of spheres

We describe all metric spaces that have sufficiently many affine functions. As an application we obtain a metric characterization of linear-convex subsets of Banach spaces.

REPRESENTATIONS WHOSE MINIMAL REDUCTION HAS A TORIC IDENTITY COMPONENT

We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally Hölder continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas Radò and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces.