arXiv research

Proves non-positivity of Hirzebruch form on stable weights and connects to flat logarithmic connections.

The study characterizes quasi Yamabe solitons with potential vector fields.

Essential self-adjointness proved for perturbed quadharmonic operators on Riemannian manifolds.

Study harmonic function growth on curved spaces, proving inequalities.

The study examines exceptional sets for radial limits of superharmonic functions on curved manifolds.

By using non-positively curved cubings of prime alternating link exteriors, we prove that certain ideal triangulations of their complements, derived from reduced alternating diagrams, are non-degenerate, in the sense that none of the edges is homotopic relative its endpoints to a peripheral arc. This guarantees that th…

Study shows some contractible complexes can't have certain immersions.

A complex disproves a curvature property.

The paper studies a relative version of non-positive immersion for 2-complex pairs and shows conditions under which a transitivity law holds.

Splitting theorem for non-positively curved Lorentzian spaces.

We consider maps into Riemannian manifolds of non-positive curvature and start developing a systematic PDE theory. We control the Sobolev $H^{2,2}$-norm of such a map in terms of its energy, the $L^2$-norm of its tension field and a topological term depending on the homotopy class. We also solve a Dirchlet problem with…

The article proves isometry theorems for specific types of manifolds.

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…

We study homogenous Weyl connections with non-positive sectional curvatures. The Cartesian product $\mathbb S^1 \times M$ carries canonical families of Weyl connections with such a property, for any Riemmanian manifold $M$. We prove that if a homogenous Weyl connection on a manifold, modeled on a unimodular Lie group, …

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

Suppose $(X,ω)$ is a compact Kähler manifold. Following Mabuchi, the space of smooth Kähler potentials $\mathcal H$ can be endowed with a Riemannian structure, which induces an infinite dimensional path length metric space $(\mathcal H,d)$. We prove that the metric completion of $(\mathcal H,d)$ can be identified with …

Paper finds non-positive Weyl connections on Lie groups, confirming a conjecture.

We prove two injectivity theorems for the geodesic ray transform on two-dimensional, complete, simply connected Riemannian manifolds with non-positive Gaussian curvature, also known as Cartan-Hadamard manifolds. The first theorem is concerned with bounded non-positive curvature and the second with decaying non-positive…

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

Study non-positive scalar curvature on hyperbolic manifolds, finding necessary and sufficient conditions.

Conformal invariants of manifolds of non-positive scalar curvature are studied in association with growth in volume and fundamental group.

Corrected and expanded a theorem about non-positively curved spaces.

Study on solvable Lie groups with specific Weyl connections.

Sharp stability estimate for tensor tomography in non-positive curvature.

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

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.

Groups with specific curvature have a regular language of geodesics.

Study on weakly G-slim complexes and non-positive immersions for group presentations.

The paper investigates subelliptic harmonic maps with potential using heat flow.

4-manifolds with non-positive curvature are essentially Euclidean.

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

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 …

We give a Pontryagin-Thom-Szucs type construction for non-positive codimensional singular maps, and obtain results about cobordism and bordism groups of -1 codimensional stable maps with prescribed singular fibers.

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

It is shown that if a $C^2$ surface $M\subset\mathbb R^3$ has negative curvature on the complement of a point $q\in M$, then the $\mathbb Z/2$-valued Poincaré-Hopf index at $q$ of either distribution of principal directions on $M-\{q\}$ is non-positive. Conversely, any non-positive half-integer arises in this fashion. …

Study finds criteria for surfaces with specific curvature properties.

In this paper by using left invariant Riemannian metrics on some 3-dimensional Lie groups we construct some complete non-Riemannian Berwald spaces of non-positive flag curvature and several families of geodesically complete locally Minkowskian spaces of zero constant flag curvature.

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.

We study biharmonic maps between Riemannian manifolds with finite energy and finite bi-energy. We show that if the domain is complete and the target of non-positive curvature, then such a map is harmonic. We then give applications to isometric immersions and horizontally conformal submersions.

Study shows non-compact convex hulls in certain metric spaces.

Maps preserve distances in non-positively curved spaces.

Harmonic maps study on surfaces with non-positive curvature.

We show that for an isometric immersion of a complete Riemannian manifold into a Riemannian manifold with non-positive curvature, the norm of the mean curvature vector field is square integrable, then it is minimal. This is a partial affirmative answer of the B. Y. Chen's conjecture.

Totally geodesic submanifolds in product spaces imply special curvature properties.

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.

Paper proves curvature estimate for curved spaces.

The paper extends Weyl formulae for Schrödinger operators with singular potentials.

In this survey article, we present some panorama of groups acting on metric spaces of non-positive curvature. We introduce the main examples and their rigidity properties , we show the links between algebraic or analytic properties of the group and geometric properties of the space. Finally, we conclude with a few conj…