Study on a weaker curvature condition for Kähler manifolds.
problem Understanding Kähler manifolds with nonpositive k-Ricci curvature. method Introducing almost nonpositive k-Ricci curvature and analyzing the twisted Kähler-Ricci flow. result Compact Kähler manifolds with almost nonpositive k-Ricci curvature have nef canonical line bundles. The study introduces a new concept of almost nonpositivity for holomorphic sectional curvature and proves its implications on nefness and Miyaoka-Yau inequalities.
problem Understanding the behavior of holomorphic sectional curvature on compact Kähler manifolds.
method Introducing a new notion of almost nonpositivity for holomorphic sectional curvature and proving its implications.
result Compact Kähler manifolds with almost nonpositive holomorphic sectional curvature have nef canonical line bundles, contain no rational curves, and satisfy Miyaoka-Yau type inequalities.
Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.
problem Understanding Weyl curvature pinching on 4-manifolds.
method Analyzing harmonic and pinched self-dual Weyl curvature, proving anti-self-duality.
result Proves anti-self-duality for compact 4-manifolds with pinched self-dual Weyl curvature.
The paper proves Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
problem Proving Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
method Analyzing maps between different classes of pseudo-Hermitian manifolds, using curvature assumptions.
result Holomorphic maps are constant under certain curvature conditions.
In a paper from 1954, Marstrand proved that if K⊂R2 with Hausdorff dimension greater than 1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions. In this article, we show that if M is a simply connected surface with non-positive curvature, then Marstrand's th…
We consider cocycles of isometries on spaces of nonpositive curvature H. We show that the supremum of the drift over all invariant ergodic probability measures equals the infimum of the displacements of continuous sections under the cocycle dynamics. In particular, if a cocycle has uniform sublinear drift, then there…
The study introduces new tensors for almost Finsler manifolds and analyzes their properties.
problem Defining and analyzing new types of Finsler manifolds.
method Introducing new Finsler manifolds, studying their properties, and deriving characteristic tensors.
result Characteristic tensors for almost Finsler manifolds have been generalized and their properties have been studied.
The measure concentration property of an mm-space X is roughly described as that any 1-Lipschitz map on X to a metric space Y is almost close to a constant map. The target space Y is called the screen. The case of Y=R is widely studied in many literature (see \cite{gromov}, \cite{ledoux}, \cite{mil2}…
The paper confirms a conjecture about the fundamental groups of ends of certain noncompact manifolds.
problem Understanding the fundamental groups of ends of noncompact manifolds with specific curvature properties.
method Analyzing the universal cover and using the concept of visibility manifolds.
result The fundamental group of each end of a manifold is almost nilpotent if its universal cover is a visibility manifold.
We use pinched smooth hyperbolization to show that every closed, nonpositively curved n-dimensional manifold M can be embedded as a totally geodesic submanifold of a closed, nonpositively curved (n+1)-dimensional manifold M^ of geometric rank one.
Nonpositive towers property in 3-manifolds spines.
problem Properties of 3-manifold spines.
method Analysis of 2-dimensional spines in aspherical 3-manifolds and 3-ball.
result Nonpositive towers property in 2-dimensional spines of aspherical 3-manifolds.
New noncompact manifolds with nonpositive curvature.
problem Creating new manifolds with specific curvature properties.
method Simple construction of manifolds with mixed ends.
result Found manifolds with fundamental groups not duality groups.
Alternative proof for static black hole uniqueness with nonpositive mass.
problem Proving static uniqueness for Kottler spacetimes with nonpositive mass.
method Alternative, more elementary proof of static uniqueness theorem.
result Alternative proof of static uniqueness for Kottler spacetimes with nonpositive mass.
New proof shows nonpositive curved manifolds have positive simplicial volume.
problem Proving positive simplicial volume for nonpositive curved manifolds.
method Using negative $([ rac{n}{4}]+1)$-Ricci curvature condition.
result Closed manifolds with nonpositive curvature have positive simplicial volume.
Prove long-time existence of pluriclosed flow on certain fibrations
problem Long-time existence of pluriclosed flow on fibrations
method General theorem on holomorphic submersions
result Long-time existence of pluriclosed flow on nilmanifolds, almost-abelian solvmanifolds, and certain complex surfaces
Positive simplicial volume proven for certain nonpositively curved manifolds.
problem Proving positive simplicial volume for specific nonpositively curved manifolds.
method Using differential geometry, proving a conjecture of Gromov in dimension three.
result Positive simplicial volume established for manifolds with certain curvature conditions.
Sharp inequalities for curved surfaces and cones.
problem Optimizing areas in nonpositively curved spaces.
method Proving inequalities for disks and triangles in cones.
result Minimal area properties for specific shapes in cones.
A 5-manifold fibers over a circle with nonpositive curvature.
problem Finding nonpositively curved manifolds that fiber over a circle.
method Exhibited a specific 5-manifold with nonpositive curvature.
result The fiber is a 4-manifold with nonhyperbolic fundamental group.
In this paper, we show that a nontrivial compact graph manifold is nonpositively curved if and only if its fundamental group virtually embeds into a right-angled Artin group. As a consequence, nonpositively curved graph manifolds have linear fundamental groups.
We show that the metric of nonpositively curved graph manifolds is determined by its geodesic flow. More precisely we show that if the geodesic flows of two nonpositively curved graph manifolds are C0 conjugate then the spaces are isometric.
Local isoperimetric inequality holds for balls with nonpositive curvature.
problem Preserving the isoperimetric ratio in perturbed ball metrics with nonpositive curvature.
method Analyzing perturbations of ball metrics with nonpositive curvature.
result Isoperimetric ratio is preserved only by homotheties of the ball.
Study classifies manifolds with nonpositive curvature and finds conformal Killing forms.
problem Characterizing manifolds with nonpositive curvature operator.
method Classification and vanishing theorems for conformal Killing forms.
result Classification and vanishing results for conformal Killing forms.
Paper defines new curvature measures for foliated manifolds and proves new theorems.
problem Estimating the diameter and splitting theorems for foliated manifolds.
method Introduced weighted mixed curvatures and new conditions to update estimates.
result Updated estimates of diameter and proved new splitting theorems.
The paper studies spectral properties of Jacobi operator for surfaces with nonpositive Euler characteristic.
problem Investigating spectral properties of the Jacobi operator for surfaces with nonpositive Euler characteristic.
method Proving a sharp upper bound for the second eigenvalue of the Jacobi operator and classifying surfaces attaining this bound.
result Totally geodesic tori maximize the second eigenvalue among compact orientable surfaces with positive genus.
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 cur…
The paper proves simplicial volume positivity for certain nonpositively curved 4-manifolds with nonzero Euler characteristic.
problem Proving positivity of simplicial volume for specific types of 4-manifolds.
method Using the Gauss-Bonnet theorem for Riemannian simplices, the paper shows that for closed nonpositively curved 4-manifolds with nonzero Euler characteristic, simplicial volume is positive.
result The paper confirms conjectures about the relationship between simplicial volume and Euler characteristic for four-dimensional manifolds.
This note is devoted to study the implications of nonpositive isotropic curvature and negative Ricci curvature for Einstein 4−Manifolds.
Study shows almost all Arnold stable solutions have no conjugate points.
problem Existence of conjugate points in Arnold stable solutions.
method Analysis of Misiołek curvature for Arnold stable solutions.
result Almost all Misiołek curvature is nonpositive for Arnold stable solutions.
We study subgroups of fundamental groups of real analytic closed 4-manifolds with nonpositive sectional curvature. In particular, we are interested in the following question: if a subgroup of the fundamental group is not virtually free abelian, does it contain a free group of rank two ? The technique involves the theor…
Finite simply connected 2-complexes with nonpositive planar curvature are collapsible.
problem Understanding the collapsibility of 2-complexes with specific curvature properties.
method Analyzing the fundamental groups and sectional curvatures of 2-complexes.
result Finite simply connected 2-complexes with nonpositive planar curvature are collapsible.
We show that the space of nonpositively curved metrics of a negatively curved manifold is highly non connected.
Study combinatorial Ricci flow on surfaces with nonpositive Euler characteristic.
problem Addressing convergence issues in Ricci flow on surfaces.
method Investigate combinatorial Ricci flow on surfaces of nonpositive Euler characteristic.
result Observation of convergence despite necessary condition not being valid.
The study characterizes Finsler metrics and proves their rigidity.
problem Characterizing and proving rigidity of Busemann convex Finsler metrics.
method Proving Finsler metrics are nonpositively curved if and only if they are affinely equivalent to a Riemannian metric of nonpositive sectional curvature.
result Finsler metrics are precisely Berwald metrics of nonpositive flag curvature.
In the context of Thurstons geometrisation program we address the question which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive curvature. We show that non-geometric Haken manifolds generically, but not always, admit such metrics. More precisely, we prove that a Haken manifold with, possibly emp…
An n-dimensional manifold M (n≥3) is called {\it generalized graph manifold} if it is glued of blocks that are trivial bundles of (n−2)-tori over compact surfaces (of negative Euler characteristic) with boundary. In this paper two obstructions for generalized graph manifold to be nonpositively curved are des…
Extends topological results for nonpositive curvature spaces.
problem Characterize and understand topological properties of BNPC spaces.
method Extends results from CAT(0) spaces to BNPC spaces, using links and geodesic bicombings.
result Locally BNPC topological manifolds have discrete singular sets and are homeomorphic to Euclidean space.
The study calculates volume and entropy asymptotics in nonpositive curvature manifolds.
problem Volume and entropy asymptotics in nonpositive curvature manifolds.
method Volume and entropy calculations using Riemannian volume and geodesic flow.
result Margulis function is continuous and constant if and only if the manifold has constant negative curvature.
We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link …
Paper investigates prescribing Chern scalar curvatures on specific manifolds.
problem Prescribing Chern scalar curvatures on noncompact Hermitian manifolds with nonpositive curvatures.
method Establishes existence results and sufficient conditions for negative curvature metrics.
result Obtains sufficient conditions for the existence of a constant negative Chern scalar curvature metric.
Loewner inequality proven for curved surfaces.
problem Proving Loewner's inequality for nonpositively curved surfaces.
method Combining Gauss-Bonnet formula with averaging argument using geodesic flow invariance.
result Found a disk with large total curvature around its center, leading to large area.
Integral inequalities for holomorphic maps prove rigidity and degeneracy theorems.
problem Rigidity and degeneracy theorems for holomorphic maps without curvature sign assumptions.
method Integral inequalities derived from holomorphic maps between complex manifolds.
result Proves rigidity and degeneracy theorems for holomorphic maps.
On R^n endowed with a riemannian metric of bounded nonpositive curvature, the weakly convex closed subsets are topologically trivial. The stability of such subsets under intersection characterizes the euclidean spaces.
New restrictions on holonomy groups for certain curvature conditions.
problem Restrictions on holonomy groups for Riemannian manifolds with specific curvature properties.
method Analyzing the curvature operator of the second kind to derive restrictions on holonomy groups.
result Holonomy groups are restricted to SO(n) or the manifold is flat for certain curvature conditions. This is a survey of topological properties of open, complete nonpositively curved manifolds which may have infinite volume. Topics include topology of ends, restrictions on the fundamental group, as well as a review of known examples.
Let (Mn,g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold Nk with c1<0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive b…
We prove that each nonpositively curved square VH-complex can be turned functorially into a locally 6-large simplicial complex of the same homotopy type. It follows that any group acting geometrically on a CAT(0) square VH-complex is systolic. In particular the product of two finitely generated free groups is systolic,…
We study algebraic conditions on a group G under which every properly discontinuous, isometric G-action on a Hadamard manifold has a G-invariant Busemann function. For such G we prove the following structure theorem: every open complete nonpositively curved Riemannian K(G,1) manifold that is homotopy equivalent to a fi…
Study of Martin boundary for rank 1 manifolds with nonpositive curvature.
problem Understanding the Martin boundary for specific types of manifolds.
method Modification of Ancona's argument to analyze the geometric and Martin boundaries.
result A residual set in the geometric boundary corresponds naturally to a subset of the Martin boundary.