Improvements of upper curvature bounds
We show that any space with a positive upper curvature bound has in a small neighborhood of any point a closely related metric with a negative upper curvature bound.
math.DG↗
arXiv subjects
Stephan Stadler
Publications and source records attributed to Stephan Stadler.
Ricci curvature in dimension 2
We prove that in two dimensions the synthetic notions of lower bounds on sectional and on Ricci curvature coincide.
math.DG↗
The structure of minimal surfaces in CAT(0) spaces
We prove that a minimal disc in a CAT(0) space is a local embedding away from a finite set of "branch points". On the way we establish several basic properties of minimal surfaces: monotonicity of area densities, density bounds, limit theorems and the existence of tangent maps. As an application, we prove Fary-Milnor's theorem in the CAT(0) setting.
math.DG↗
Conformal deformations of CAT(0) spaces
We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.
math.DG↗
Affine functions on Alexandrov spaces
We show that every finite-dimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.
math.MG↗
An obstruction to the smoothability of singular nonpositively curved metrics on 4-manifolds by patterns of incompressible tori
We give new examples of closed smooth 4-manifolds which support singular metrics of nonpositive curvature, but no smooth ones, thereby answering affirmatively a question of Gromov. The obstruction comes from patterns of incompressible 2-tori sufficiently complicated to force branching of geodesics for nonpositively curved metrics.
math.MG↗