Selberg's Lemma fails for certain curved manifolds.
problem Applying Selberg's Lemma to negatively curved Hadamard manifolds.
method Proving the failure of Selberg's Lemma for specific groups of manifolds.
result Selberg's Lemma does not hold for discrete isometry groups of negatively curved Hadamard manifolds.
We state and prove a Chern-Osserman-type inequality in terms of the volume growth for complete surfaces with controlled mean curvature properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity KN≤b<0
Study on minimal submanifolds in curved spaces with unique solution to asymptotic Plateau problem.
problem Minimal submanifolds in negatively curved spaces with small curvature.
method Analysis of spheres at infinity and asymptotic Plateau problem.
result Complete minimal submanifolds bound a class of spheres with uniquely solvable asymptotic Plateau problem.
Study finds conditions for global minimizers on curved manifolds with fast diffusion and nonlocal interactions.
problem Existence of global minimizers for a free energy functional on negatively curved manifolds.
method Investigation of Carlson-Levin type inequalities for Cartan-Hadamard manifolds.
result Establishes necessary and sufficient conditions for the existence of global energy minimizers.
Let X, Y be the universal covers of two compact Riemannian manifolds (with dimension not equal to 4) with negative sectional curvature. Then every quasiisometry between them lies at a finite distance from a bilipschitz homeomorphism.
We prove that a quasi-isometric map, and more generally a coarse embedding, between pinched Hadamard manifolds is within bounded distance from a unique harmonic map.
In this paper, we prove a version of the classical Cartan-Hadamard theorem for negatively curved manifolds, of dimension n=5, with non-empty totally geodesic boundary. More precisely, if M1n,M2n are any two such manifolds, we show that (1) ∂∞M~1n is homeomorphic to $\partial ^\infty…
Estimates Poisson kernel on negatively curved Hadamard manifolds.
problem Estimating the Poisson kernel on Hadamard manifolds with negative curvature.
method Using techniques from Anderson-Schoen for estimating positive harmonic functions in cones.
result Global upper and lower bounds for the Poisson kernel are derived.
One-harmonic maps from a curved surface to hyperbolic plane have specific interior properties.
problem Characterizing one-harmonic maps from curved surfaces to hyperbolic spaces.
method Using Minkowski geometry and interpreting maps as Gauss maps of convex surfaces.
result One-harmonic maps have images confined to the interior of convex hulls.
The study examines the normal growth exponent of submanifolds in negatively curved manifolds.
problem Understanding the normal growth exponent of submanifolds in negatively curved manifolds.
method Analyzing the geodesic flow and operator norms on submanifolds bi-Lipschitz to hyperbolic spaces.
result If a submanifold's normal growth exponent is at most 1, the ambient manifold is bi-Lipschitz to hyperbolic space.
We state and prove a Chern-Osserman Inequality in terms of the volume growth for minimal surfaces properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity.
We study the long-time behaviour of nonnegative solutions of the Porous Medium Equation posed on Cartan-Hadamard manifolds having very large negative curvature, more precisely when the sectional or Ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole. We find an une…
Constructs harmonic maps near retractions in hyperbolic spaces.
problem Finding harmonic maps near retractions in hyperbolic spaces.
method Constructs harmonic maps to the hyperbolic plane from quasidisks, and to convex hulls from sets in the boundary at infinity of pinched Hadamard manifolds.
result Harmonic maps are bounded from nearest-point retractions in hyperbolic spaces.
Linear inequality found for certain curved spaces.
problem Finding isoperimetric inequalities for curved spaces.
method Extending known result to homogeneous Hadamard manifolds.
result Linear isoperimetric inequality proven for higher-dimensional cycles.
The paper examines functional properties on manifolds with very negative curvature.
problem Functional properties on manifolds with very negative curvature.
method New Hardy-type inequalities and first and second order inequalities.
result Functional properties typically hold in manifolds with polynomially growing negative curvature.
Closed surfaces minimize total curvature in curved spaces.
problem Minimizing total curvature in curved spaces.
method Isometric embedding via holonomy and Pogorelov's theory.
result Closed surfaces bound flat convex bodies.
Sharp Minkowski inequality for convex surfaces in curved spaces.
problem Establishing a precise lower bound for total mean curvature of convex surfaces.
method Harmonic mean curvature flow applied to Cartan-Hadamard manifolds.
result Improved Minkowski inequality for convex surfaces in nonpositively curved 3-spaces.
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.
We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, C) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the assumption of bounded torsion, to the case of negatively pinched Hadamard manifolds. Eve…
In this paper, we generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, C) to geometrically infinite discrete subgroups Γ of isometries of negatively pinched Hadamard manifolds X. We then generalize a theorem of Bishop to prove that every discrete geome…
Study counts surface subgroups in curved 3D manifolds.
problem Count surface subgroups in curved 3D manifolds.
method Solve foliated Plateau problem in Cartan-Hadamard manifolds.
result Prove rigidity for lower bound on surface subgroup count.
In this article we prove upper bounds for the Laplace eigenvalues λk below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of k2 and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose section…
The study extends isoperimetric inequalities to non-positive curvature spaces.
problem Isoperimetric inequalities in spaces of non-positive curvature.
method Analyzes submanifolds and geodesics in Cartan-Hadamard manifolds.
result Extensions of isoperimetric inequalities to non-positive curvature spaces.
Groups can act on Lp spaces with non-trivial cohomology.
problem Understanding actions of groups on Hadamard spaces.
method Analyzing isometric actions on Hadamard manifolds and Lp-spaces. result Non-trivial Lp-cohomology for certain groups. New entropy functionals for curved spaces help predict shape behavior.
problem Understanding entropy behavior in curved spaces.
method Introduced new entropy functionals for submanifolds of Cartan-Hadamard manifolds.
result Obtained sharp lower bounds on these entropies for certain closed hypersurfaces and observed a novel rigidity phenomenon.
Geometrically proves majorizing measure theorem on Hadamard manifolds.
problem Volume size relation between random process index space and its convex hull.
method Assumed Hadamard manifold, derived upper bound for volume ratio, applied to prove majorizing measure theorem.
result Upper bound for volume ratio between index space and convex hull.
Study finds solitons on curved spaces with varying behavior.
problem Existence and behavior of solitons on curved spaces.
method Proved existence of entire graphical translators on Cartan-Hadamard manifolds, analyzed asymptotic behavior based on curvature.
result Asymptotic behavior of solitons depends on curvature; bounded solutions exist under certain conditions.
Formula for comparing total curvature in curved spaces, with applications to isoperimetric problems.
problem Comparing total curvature in curved spaces and solving isoperimetric problems.
method Explicit formula for total curvature of level sets, applications to isoperimetric problems.
result Explicit formula for comparing total curvature in Riemannian manifolds.
This paper derives radial fields on manifolds of symmetric positive definite matrices.
problem Lack of an expression for radial fields on manifolds of symmetric positive definite matrices.
method Derives an expression for radial fields on manifolds of symmetric positive definite matrices.
result Derives an expression for radial fields on manifolds of symmetric positive definite matrices.
We discuss a variation of Gromov's notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. The Nagata dimension turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with finite Nagata dimension includes in particular all doubling spaces, metric trees, …
Estimates prove existence of curvature flow in curved spaces.
problem Mean curvature flow in curved spaces with boundary conditions.
method A priori estimates and existence proof for curvature flow.
result Existence of curvature flow with asymptotic Dirichlet conditions.
Convex hypersurfaces in curved spaces bound convex regions.
problem Characterizing convex hypersurfaces in curved spaces.
method Gauss-Codazzi equations, Schur comparison theorem, Alexandrov geometry.
result Closed convex hypersurfaces bound convex regions in curved spaces.
The paper proves Liouville theorems for V-harmonic maps under specific curvature conditions.
problem Proving Liouville theorems for V-harmonic maps in Riemannian manifolds with non-negative (m,V)-Ricci curvature. method Probabilistic proof extending previous results by Cheng, Hildebrandt-Jost-Wideman, and Stafford.
result Extends Liouville theorems to a broader class of manifolds and curvature conditions.
We consider the porous medium equation with power-type reaction terms up on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If p>m, small data give rise to global-in-time solutions while solutions associated to large data blow up in finite t…
Study on existence of ground states on curved spaces with conditions on potential growth.
problem Existence of ground states for aggregation-diffusion models on Cartan-Hadamard manifolds.
method Investigation of a free energy functional on Cartan-Hadamard manifolds, considering entropy and interaction energies.
result Necessary and sufficient conditions for existence of ground states are found, depending on the growth of the attractive potential.
Generalizing results due to Brady and Farb we prove the existence of a bilipschitz embedded manifold of pinched negative curvature and dimension m_1+m_2-1 in the product X:=X_1^{m_1} times X_2^{m_2} of two Hadamard manifolds X_i^{m_i} of dimension m_i with pinched negative curvature. Combining this result with a Theore…
Geometric approach to Dirac operator evolution on spacetimes.
problem Constructing the Cauchy evolution operator for Lorentzian Dirac operators.
method Realizing the operator as a sum of oscillatory integrals, relating to Feynman propagator.
result Relating Cauchy evolution operators to Feynman propagators and constructing Hadamard states.
A new depth measure and median defined on Hadamard manifolds.
problem Statistical depth and median on Hadamard manifolds.
method Horospherical depth and Busemann median defined using renormalized distance functions.
result The Busemann median exists for every Borel probability measure on Hadamard manifolds.
3-manifolds with convex boundary are rigid in certain curvature conditions.
problem Characterizing the rigidity of nonpositively curved manifolds with convex boundaries.
method Using a comparison formula for total curvature of Riemannian hypersurfaces, the authors prove the rigidity of the manifolds.
result Compact Riemannian 3-manifolds with strictly convex simply connected boundary and sectional curvature K≤a≤0 are isometric to a convex domain in a complete simply connected space of constant curvature a.
Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.
problem Preserving convexity and convergence of curves under curvature flows on pinched Hadamard surfaces.
method Area- and length-preserving curvature flows, refined comparison arguments, delicate curvature estimates.
result Convexity is preserved and curves converge to a geodesic circle under certain conditions.
Study investigates non-existence of bounded solutions on curved spaces.
problem Non-existence of bounded solutions to semi-linear elliptic equations on Cartan-Hadamard manifolds.
method Novel comparison technique using convex hypersurfaces.
result Extends previous results to curved spaces, highlighting curvature's role.
Paper proves uniqueness of minimal maps in curved spaces.
problem Proving uniqueness of minimal maps into Cartan-Hadamard manifolds.
method Proof based on convexity of functions in terms of squared singular values.
result Uniqueness theorem for minimal maps into Riemannian manifolds.
Study extends convexity in curved spaces using fractional integrals.
problem Extending convexity to curved spaces with nonpositive curvature.
method Introducing (geodesically) h-convex functions and using Katugampola's fractional integrals. result Essentially sharp estimate involving squared distance mappings.
In the first part of the paper we investigate some geometric features of Moser-Trudinger inequalities on complete non-compact Riemannian manifolds. By exploring rearrangement arguments, isoperimetric estimates, and gluing local uniform estimates via Gromov's covering lemma, we provide a Coulhon, Saloff-Coste and Varopo…
The paper defines subdifferentials on Hadamard manifolds and identifies conditions for Fenchel conjugate equality.
problem Understanding convex analysis on Riemannian manifolds.
method Using Busemann functions to define subdifferentials and investigate Fenchel conjugate equality.
result Identifies conditions for equality in the Fenchel-Young inequality on Hadamard manifolds.
New harmonic Hadamard manifolds defined via hypergeometric equations.
problem Characterizing harmonic Hadamard manifolds using hypergeometric equations.
method Defining harmonic Hadamard manifolds of hypergeometric type and using spherical Fourier transform.
result Characterization of harmonic Hadamard manifolds being of hypergeometric type.
Solutions found for specific curvature problems in geometric settings.
problem Finding hypersurfaces with prescribed mean curvature in geometric settings.
method Proving existence of solutions to the asymptotic Plateau problem in Cartan-Hadamard manifolds.
result Existence of solutions to the asymptotic Plateau problem for hypersurfaces of prescribed mean curvature.
We study asymptotically harmonic manifolds of negative curvature, without any cocompactness or homogeneity assumption. We show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces: in particular, we determine the volume entropy, the spectrum and the relative densities of …