arXiv research

We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and Bruhat--Tits buildings. Applications to discrete groups and further dev…

Splitting theorem for non-positively curved Lorentzian spaces.

Paper proves curvature estimate for curved spaces.

We characterize symmetric spaces of non-positive curvature by the equality case of general inequalities between geometric quantities

The paper proves rigidity results for non-positively curved homogeneous Finsler metrics.

Maps preserve distances in non-positively curved spaces.

Totally geodesic submanifolds in product spaces imply special curvature properties.

We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved and CAT(-1) metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this and other formulas in terms of curvature, we prove an analogue of the Eels-Sampson…

Defines timelike ideal boundary for non-positively curved Lorentzian spaces.

We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residua…

In this paper, we investigate the geometry of the moduli space of curves by using the curvature properties of direct image sheaves of vector bundles. We show that the moduli space $(M_g, ω_{WP})$ of curves with genus $g>1$ has dual-Nakano negative and semi-Nakano-negative curvature, and in particular, it has non-positi…

We show that a space with a finite asymptotic dimension is embeddable in a non-positively curved manifold. Then we prove that if a uniformly contractible manifold X is uniformly embeddable in $\R^n$ or non-positively curved n-dimensional simply connected manifold then $X\times\R^n$ is integrally hyperspherical. If a un…

The generalized Chen's conjecture on biharmonic submanifolds asserts that any biharmonic submanifold of a non-positively curved manifold is minimal (see e.g., [CMO1], [MO], [BMO1], [BMO2], [BMO3], [Ba1], [Ba2], [Ou1], [Ou2], [IIU]). In this paper, we prove that this conjecture is false by constructing foliations of pro…

Study harmonic function growth on curved spaces, proving inequalities.

If a class of finitely generated groups Curly(G) is closed under isometric amalgamations along free subgroups, then every G in Curly(G) can be quasi-isometrically embedded in a group Hat(G) in Curly(G) that has no proper subgroups of finite index. Every compact, connected, non-positively curved space X admits an isomet…

Survey on harmonic maps in non-smooth spaces, focusing on rigidity.

The paper proves geometrical finiteness for automorphism groups of K3 surfaces and related varieties.

S. Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. We solve the geodesic equation and its pert…

The study shows that ergodic measures are not generic on non-positively curved manifolds.

Groups with specific curvature have a regular language of geodesics.

The paper extends a prediction method to curved spaces.

Survey on non-positively curved cube complexes and geometric group theory.

Linear inequality found for certain curved spaces.

Fold maps associated to geodesic random walks on curved spaces.

We prove some rigidity results on geodesic orbit Finsler spaces with non-positive curvature. In particular, we show that a geodesic Finsler space with strictly negative flag curvature must be a non-compact Riemannian symmetric space of rank one.

We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the …

This article generalizes the work of Ballmann and Światkowski to the case of Reflexive Banach spaces and uniformly convex Busemann spaces, thus giving a new fixed point criterion for groups acting on simplicial complexes.

For a k-flat F inside a locally compact CAT(0)-space X, we identify various conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our conditions are formulated in terms of the ultralimit of X. As applications, we obtain (1) constraints on the behavior of quasi-isometries between tocally compact CAT(0…

Using the Perron method, we prove the existence of hypersurfaces of prescribed special Lagrangian curvature with prescribed boundary inside complete Riemannian manifolds of non-positive curvature.

We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero curvature bounds.

Free finite group actions on non-positively curved 3-manifolds

Study approximate marked length spectrum rigidity in non-positively curved groups.

Equality in Miyaoka-Yau inequality implies uniformization of Klt pairs.

We give conceptual proofs of some well known results concerning compact non-positively curved locally symmetric spaces. We discuss vanishing and non-vanishing of Pontrjagin numbers and Euler characteristics for these locally symmetric spaces. We also establish vanishing results for Stiefel-Whitney numbers of (finite co…

We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard manifolds are established for singular spaces; new arithmeticity and rigidity state…

We deform a map into a Riemannian manifold that is horizontal with respect to a submersion onto a non-positively curved manifold and satisfies a Chow condition into a harmonic one through a horizontal homotopy.

Generalizes Leighton's theorem to cube complexes.

We prove that rationally essential manifolds with suitably large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds of positive dimension. Particular examples include all manifolds of non-positive sectional curvature of rank one and all irreducible locally symmetric spaces of …

We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a flat spacetime of dimension (2+1). The proof follows from polyhedral approximation.

Spaces of circle embeddings in curved surfaces indexed by trees.

Proper proximality proved for various groups on non-positive curvature spaces.

New conditions ensure surface group extensions are non-positively curved.

We extend to any simply connected Kähler manifold with non-positive sectional curvature some conditions for interpolation in $\mathbb{C}$ and in the unit disk given by Berndtsson, Ortega-Cerdà and Seip. The main tool is a comparison theorem for the Hessian in Kähler geometry due to Greene, Wu and Siu, Yau.

We amend the statement of point~(i) in Theorem~1.3 in arxiv:0901.1022 and supply the additional arguments and minor changes for the results that depend on it. We also seize the occasion and generalize to non-finitely generated lattices.

In this article we give an explicit algorithm which will determine, in a discrete and computable way, whether a finite piecewise Euclidean complex is non-positively curved. In particular, given such a complex we show how to define a boolean combination of polynomial equations and inequalities in real variables, i.e. a …

Billiard trajectories in curved spaces have predictable travel times.

This paper concerns a study of three families of non-compact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show they have finite telescopic dimension. We also show the existence of Furstenberg maps for some group actions on these spaces. Suc…

Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space W(X). In this paper we investigate the geometry of W(X) when X is a Hadamard space, by which we mean that $X$ has globally non-positi…