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Stephan Stadler

Publications and source records attributed to Stephan Stadler.

At least 19 recordsLinked to original sources
arXiv2025

Isoperimetric inequalities in Hadamard spaces of asymptotic rank two

Gromov's isoperimetric gap conjecture for Hadamard spaces states that cycles in dimensions greater than or equal to the asymptotic rank admit linear isoperimetric filling inequalities, as opposed to the inequalities of Euclidean type in lower dimensions. In the case of asymptotic rank 2, recent progress was made by Dru\c{t}u-Lang-Papasoglu-Stadler who established a homotopical inequality for Lipschitz 2-spheres with exponents arbitrarily close to 1. We prove a homological inequality of the same type for general cycles in dimensions at least 2, assuming that the ambient space has finite linearly controlled asymptotic dimension. This holds in particular for all Hadamard 3-manifolds and finite-dimensional CAT(0) cube complexes.

math.MG↗
arXiv2025

Minimal tetrahedra and an isoperimetric gap theorem in non-positive curvature

We investigate isoperimetric inequalities for Lipschitz 2-spheres in CAT(0) spaces, proving bounds on the volume of efficient null-homotopies. In one dimension lower, it is known that a quadratic inequality with a constant smaller than $c_2=1/(4\pi)$ -- the optimal constant for the Euclidean plane -- implies that the underlying space is Gromov hyperbolic, and a linear inequality holds. We establish the first analogous gap theorem in higher dimensions: if a proper CAT(0) space satisfies a Euclidean inequality for 2-spheres with a constant below the sharp threshold $c_3=1/(6\sqrt{\pi})$, then the space also admits an inequality with an exponent arbitrarily close to 1. As a corollary we obtain a similar result for Lipschitz surfaces of higher genus. Towards our main theorem we prove a (non-sharp) Euclidean isoperimetric inequality for null-homotopies of 2-spheres, apparently missing in the literature. A novelty in our approach is the introduction of minimal tetrahedra, which we demonstrate satisfy a linear inequality.

math.MG↗
arXiv2024

Rigidity of ideal symmetric sets

We characterize higher rank model geometries -- Riemannian symmetric spaces, Euclidean buildings and products -- among Hadamard spaces by using antipodal sets at infinity.

math.MG↗
arXiv2023

Isoperimetric inequalities vs. upper curvature bounds

The Dehn function of a metric space measures the area necessary in order to fill a closed curve of controlled length by a disc. As a main result, we prove that a length space has curvature bounded above by $\kappa$ in the sense of Alexandrov if and only if its Dehn function is bounded above by the Dehn function of the model surface of constant curvature $\kappa$. This extends work of Lytchak and the second author from locally compact spaces to the general case. A key ingredient in the proof is the construction of minimal discs with suitable properties in certain ultralimits. Our arguments also yield quantitative local and stable versions of our main result. The latter has implications on the geometry of asymptotic cones.

math.DG↗
arXiv2022

CAT(0) spaces of higher rank I

A CAT(0) space has rank at least $n$ if every geodesic lies in an $n$-flat. Ballmann's Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least $2$ with a geometric group action is rigid -- isometric to a Riemannian symmetric space, a Euclidean building, or splits as a metric product. This paper is the first in a series motivated by Ballmann's conjecture. Here we prove that a CAT(0) space of rank at least $n\geq 2$ is rigid if it contains a periodic $n$-flat and its Tits boundary has dimension $(n-1)$. This does not require a geometric group action. The result relies essentially on the study of flats which do not bound flat half-spaces -- so-called Morse flats. We show that the Tits boundary $\partial_T F$ of a periodic Morse $n$-flat $F$ contains a regular point -- a point with a Tits-neighborhood entirely contained in $\partial_T F$. More precisely, we show that the set of singular points in $\partial_T F$ can be covered by finitely many round spheres of positive codimension.

math.MG↗
arXiv2022

CAT(0) spaces of higher rank II

This belongs to a series of papers motivated by Ballmann's Higher Rank Rigidity Conjecture. We prove the following. Let $X$ be a CAT(0) space with a geometric group action. Suppose that every geodesic in $X$ lies in an $n$-flat, $n\geq 2$. If $X$ contains a periodic $n$-flat which does not bound a flat $(n+1)$-half-space, then $X$ is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a metric product. This generalizes the Higher Rank Rigidity Theorem for Hadamard manifolds with geometric group actions.

math.MG↗
arXiv2022

Rank Rigidity for CAT(0) spaces without 3-flats

If a group $\Gamma$ acts geometrically on a CAT(0) space $X$ without 3-flats, then either $X$ contains a $\Gamma$-periodic geodesic which does not bound a flat half-plane, or else $X$ is a rank 2 Riemannian symmetric space, a 2-dimensional Euclidean building or non-trivially splits as a metric product. Consequently all such groups satisfy a strong form of the Tits Alternative.

math.MG↗
arXiv2022

CAT(0) spaces of higher rank

Ballmann's Rank Rigidity Conjecture predicts that a CAT(0) space of higher rank with a geometric group action is rigid -- isometric to a Riemannian symmetric space, a Euclidean building, or splits as a direct product. We confirm this conjecture in rank 2. For CAT(0) spaces of higher rank $n\geq 2$ we prove rigidity if the space contains a periodic $n$-flat and every complete geodesic lies in some $n$-flat. This does not require a geometric group action.

math.MG↗
arXiv2021

Curvature bounds of subsets in dimension two

We show that closed subsets with vanishing first homology in two-dimensional spaces inherit the upper curvature bound from their ambient spaces and discuss topological applications.

math.DG↗
arXiv2020

Morse Quasiflats II

This is the second in a two part series of papers concerning Morse quasiflats - higher dimensional analogs of Morse quasigeodesics. Our focus here is on their asymptotic structure. In metric spaces with convex geodesic bicombings, we prove asymptotic conicality, uniqueness of tangent cones at infinity and Euclidean volume growth rigidity for Morse quasiflats. Moreover, we provide some immediate consequences.

math.MG↗
arXiv2019

Morse Quasiflats I

This is the first in a series of papers concerned with Morse quasiflats, which are a generalization of Morse quasigeodesics to arbitrary dimension. In this paper we introduce a number of alternative definitions, and under appropriate assumptions on the ambient space we show that they are equivalent and quasi-isometry invariant; we also give a variety of examples. The second paper proves that Morse quasiflats are asymptotically conical and have canonically defined Tits boundaries; it also gives some first applications.

math.MG↗
arXiv2019

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↗
arXiv2018

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↗
arXiv2017

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↗
arXiv2017

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↗