Study shows some contractible complexes can't have certain immersions.
problem Understanding non-positive immersions in contractible complexes.
method Provided counterexamples to a conjecture by Wise.
result Some contractible complexes do not have non-positive immersions.
A complex disproves a curvature property.
problem Proving non-equivalence of curvature properties.
method Analyzing a specific 2-complex presentation.
result The contracting non-positive immersions property is not equivalent to positive maximal irreducible curvature.
The paper studies a relative version of non-positive immersion for 2-complex pairs and shows conditions under which a transitivity law holds.
problem The study of collapsing non-positive immersion for 2-complex pairs and its implications.
method Introduced a relative version of collapsing non-positive immersion for 2-complex pairs (L,K) and proved a transitivity law under certain conditions. result Under certain conditions, a transitivity law holds: If (L,K) has relative collapsing non-positive immersion and K has collapsing non-positive immersion, then L has collapsing non-positive immersion. Splitting theorem for non-positively curved Lorentzian spaces.
problem Understanding curvature in Lorentzian spaces.
method Proving a splitting theorem with global non-positive timelike curvature and extending first variation formula.
result Splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature.
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 S1×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, …
Non-positively curved surfaces can be embedded in flat spacetime.
problem Embedding surfaces with non-positive curvature in flat spacetime.
method Polyhedral approximation to prove isometric embedding.
result Metric of non-positive curvature can be embedded as a convex spacelike Cauchy surface.
The paper proves rigidity results for non-positively curved homogeneous Finsler metrics.
problem Rigidity of non-positively curved homogeneous Finsler metrics.
method Analyzes and proves rigidity results for specific Finsler metrics with non-positive flag curvature.
result Homogeneous Finsler spaces with non-positive flag curvature and isotropic S-curvature are either Riemannian or locally Minkowskian.
Paper finds non-positive Weyl connections on Lie groups, confirming a conjecture.
problem Finding non-positive invariant Weyl connections on Lie groups.
method Investigation of completely solvable Lie groups, focusing on SOL group.
result Only SOL admits non-positive Weyl connections, confirming a conjecture.
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.
problem Finding metrics with non-positive scalar curvature on asymptotically hyperbolic manifolds.
method Analyzing the prescribed scalar curvature problem for non-positive scalar curvature on asymptotically hyperbolic manifolds.
result Obtained a necessary and sufficient condition for the existence of solutions.
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.
problem Correcting and expanding a theorem about non-positively curved spaces.
method Amended and generalized a theorem about non-positively curved spaces.
result Corrected and generalized a theorem about non-positively curved spaces.
Study on solvable Lie groups with specific Weyl connections.
problem Characterizing solvable Lie groups with invariant stretched non-positive Weyl connections.
method Analyzing structure and classification of solvable Lie groups.
result Classification of solvable Lie groups and compact solvmanifolds with invariant SNP connections.
Paper examines non-positive curvature properties of Hilbert metric in convex domains.
problem Investigating curvature properties of Hilbert metric in convex domains.
method Surveying and proving relationships among concepts, showing conditions for rigidity.
result If Hilbert metric is Berwald, domain is an ellipsoid and metric is Riemannian.
The study proves local and global Holder continuity of quasi-n-harmonic mappings into spaces with non-positive curvature.
problem Holder continuity of quasi-n-harmonic mappings into metric spaces with non-positive curvature.
method Local and global Holder continuity proved using Euclidean domains and bounded Lipschitz domains.
result Local and global Holder continuity of quasi-n-harmonic mappings.
Sharp stability estimate for tensor tomography in non-positive curvature.
problem Stability estimate for tensor tomography on manifolds with non-positive curvature.
method Pestov identity with localized frequency boundary term.
result Stability estimate of the form L2↦HT1/2. Injectivity theorems for geodesic ray transform on curved 2D manifolds.
problem Understanding geodesic ray transform on curved 2D manifolds.
method Injectivity theorems for geodesic ray transform on Cartan-Hadamard manifolds with non-positive curvature.
result Proved two injectivity theorems for geodesic ray transform on 2D Cartan-Hadamard manifolds.
The study shows that ergodic measures are not generic on non-positively curved manifolds.
problem Determining the genericity of ergodic measures on non-positively curved Riemannian manifolds.
method Investigates the existence of an open isometric embedding of a product manifold with a factor isometric to S1. result The closure of the set of ergodic measures does not encompass all invariant measures, indicating the failure of genericity.
Groups with specific curvature have a regular language of geodesics.
problem Understanding the language of geodesics in non-positively curved triangle groups.
method Proving finitely many cone types and regularity of geodesic languages.
result The language of lexicographically first geodesics is regular and satisfies the fellow traveller property.
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.
Study on weakly G-slim complexes and non-positive immersions for group presentations.
problem Conditions for non-positive immersions in group presentations.
method Investigation of weakly G-slim complexes and their relationship to left-orderable groups.
result Conditions on generalized Wirtinger presentations guaranteeing non-positive immersions for their associated 2-complexes.
4-manifolds with non-positive curvature are essentially Euclidean.
problem Understanding the structure of 4-manifolds with specific curvature properties.
method Proving homeomorphism to Euclidean space using globally non-positive curvature.
result CAT(0) 4-manifolds are homeomorphic to Euclidean space.
The paper examines complete Yamabe solitons with finite total scalar curvature.
problem Characterizing complete Yamabe solitons with specific curvature properties.
method Analyzing steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature.
result Steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive scalar curvature have zero scalar curvature.
Survey on harmonic maps in non-smooth spaces, focusing on rigidity.
problem Rigidity phenomena in non-smooth spaces.
method Regularity theory of harmonic maps to non-smooth targets.
result Generalizations of Margulis superrigidity and holomorphic rigidity of Teichmüller space.
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.
problem None explicitly stated, focuses on introduction.
method Lecture notes and mini-course teaching.
result Introduction to non-positively curved cube complexes and geometric group theory.
It is shown that if a C2 surface M⊂R3 has negative curvature on the complement of a point q∈M, then the 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.
problem Understanding Kählerian or projective structures on surfaces with non-positive curvature.
method Established a criterion for compact Hermitian surfaces with non-positive second Chern-Ricci curvature.
result Found conditions for Kählerian or projective structures on surfaces with non-positive curvature.
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 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.
problem Compactness of convex hulls in weakly non-positive curvature spaces.
method Introduced a conical geodesic bicombing and used it to construct a counterexample.
result Existence of a metric space with a finite subset whose convex hull is not compact.
Maps preserve distances in non-positively curved spaces.
problem Understanding continuous maps between boundaries of Hadamard manifolds.
method Circumcenter extension maps, cross ratio preservation, visibility conditions.
result Circumcenter extension maps are rough isometries under certain conditions.
Graphs with some negative Bakry-Émery curvature have explicit diameter bounds.
problem Understanding graphs with non-constant Bakry-Émery curvature.
method Proving distance bounds for graphs with positive Bakry-Émery curvature except for a finite or infinite set of non-positively curved vertices.
result Explicit upper bounds for the diameter of graphs with non-constant Bakry-Émery curvature.
Harmonic maps study on surfaces with non-positive curvature.
problem Characterize harmonic maps on surfaces with non-positive curvature.
method Transversality theory for Banach manifolds.
result Set of somewhere injective harmonic maps is open, dense, and connected.
Totally geodesic submanifolds in product spaces imply special curvature properties.
problem Characterizing manifolds based on the existence of totally geodesic submanifolds.
method Analyzing totally geodesic submanifolds in products of non-positively curved manifolds with transversality conditions.
result If infinitely or just a single dense such submanifolds exist, the manifold must be locally symmetric.
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.
We extend to any simply connected Kähler manifold with non-positive sectional curvature some conditions for interpolation in 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.
problem Estimating total curvature of closed hypersurfaces.
method Used non-positively curved symmetric spaces.
result Proved total curvature estimate and isoperimetric inequality.
Paper proves non-positivity of eigenvalue for Schrödinger operator on hypersurfaces.
problem Eigenvalue non-positivity of Schrödinger operator on hypersurfaces.
method Penalizes (r+1)-mean curvature, establishes non-positivity. result Characterizes the sphere when eigenvalue is null.
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…
In this paper two metric properties on geodesic length spaces are introduced by means of the metric projection, studying their validity on Alexandrov and Busemann NPC spaces. In particular, we prove that both properties characterize the non-positivity of the sectional curvature on Riemannian manifolds. Further results …
Improved spectral projection estimates on manifolds of non-positive curvature.
problem Estimating spectral projections on manifolds with non-positive curvature.
method New spectral projection estimates, including sharp ones for tori, using pointwise estimates and microlocal L2oLqc Kakeya-Nikodym estimates. result Stronger and more precise spectral projection estimates, including new sharp estimates for tori.
In this note we prove that any closed graph manifold admitting a metric of non-positive sectional curvature (NPC-metric) has a finite cover, which is fibered over the circle. An explicit criterion to have a finite cover, which is fibered over the circle, is presented for the graph manifolds of certain class.
No conformal product structures on compact manifolds with constant curvature.
problem Existence of conformal product structures on compact manifolds with constant curvature.
method Analyzing non-flat manifolds and irreducible, compact locally symmetric spaces of non-positive curvature.
result Compact non-flat manifolds with constant sectional curvature admit no conformal product structure.
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…
Metric spaces with non-positive curvature have a specific curve isoperimetric property.
problem Understanding spaces with non-positive curvature.
method Proving isoperimetric inequality for metric spaces with Alexandrov curvature.
result Metric spaces with non-positive curvature have a specific curve isoperimetric property.
Free finite group actions on non-positively curved 3-manifolds
problem Resolution of the remaining case of graph manifolds
method Proving the existence of a G-invariant NPC metric result Resolution of the remaining case of graph manifolds