Study shows spectrum properties for specific Hadamard manifolds.
problem Spectrum properties of Hadamard manifolds.
method Absolute continuity and spectrum determination for two classes of Hadamard manifolds.
result Spectrum properties determined for specific Hadamard manifolds.
Sharp curvature inequality extends Euclidean isoperimetric inequality to Cartan-Hadamard manifolds.
problem Proving the Euclidean isoperimetric inequality on Cartan-Hadamard manifolds with nullity.
method Using the Chern-Gauss-Bonnet theorem to establish a sharp inequality for total curvature.
result The Euclidean isoperimetric inequality extends to Cartan-Hadamard manifolds with nullity.
A new class of harmonic Hadamard manifolds, those spaces called of hypergeometric type, is defined in terms of Gauss hypergeometric equations. Spherical Fourier transform defined on a harmonic Hadamard manifold of hypergeometric type admits an inversion formula. A characterization of harmonic Hadamard manifold being of…
Solves Plateau problem for surfaces in pinched curvature manifolds.
problem Asymptotic Plateau problem for immersed surfaces in pinched curvature manifolds.
method Complete solution to asymptotic Plateau problem, providing dynamical stability of hypersurface laminations.
result Achieved complete solution to the asymptotic Plateau problem for immersed surfaces of constant extrinsic curvature in Cartan--Hadamard manifolds.
Various results based on some convexity assumptions (involving the exponential map along with affine maps, geodesics and convex hulls) have been recently established on Hadamard manifolds. In this paper we prove that these conditions are mutually equivalent and they hold if and only if the Hadamard manifold is isometri…
The paper proves Liouville-type theorems on Hadamard manifolds.
problem Non-existence of Killing-Yano tensors, Killing tensors, and harmonic symmetric tensors on Hadamard manifolds.
method Proofs use Liouville-type theorems on non-existence of subharmonic and harmonic functions on complete Riemannian manifolds, modified for Hadamard manifolds.
result Proves several Liouville-type theorems on Hadamard manifolds.
Maps between Hadamard manifolds are quasi-isometric to harmonic maps.
problem Understanding the relationship between quasi-isometric maps and harmonic maps on Hadamard manifolds.
method Extending a previous result to quotient spaces of Hadamard manifolds by convex cocompact discrete groups.
result Locally quasi-isometric maps to Hadamard manifolds are within bounded distance from a unique harmonic map.
Upper bounds for circumradius in Hadamard surfaces with curvature constraints.
problem Bounding circumradius in Hadamard surfaces with curvature constraints.
method Using curvature constraints to derive upper bounds for circumradius.
result Upper bounds for circumradius in terms of curvature bounds.
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.
The paper defines Fenchel conjugate and biconjugate on Hadamard manifolds.
problem Defining Fenchel conjugate and biconjugate on curved spaces.
method Introduced a new definition of Fenchel conjugate and biconjugate on Hadamard manifolds based on the tangent bundle.
result Developed a Fenchel-Moreau Theorem for geodesically convex functions on Hadamard manifolds.
Nonexistence of radial optimal functions on certain Cartan-Hadamard manifolds.
problem Proving nonexistence of radial optimal functions for the Sobolev inequality on Cartan-Hadamard manifolds.
method Ad hoc arguments not relying on the Cartan-Hadamard conjecture.
result If the optimal constant in the Sobolev inequality is achieved by a radial function, then the manifold must be isometric to Euclidean space.
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.
Study KKT conditions for multi-objective optimization on Hadamard manifolds.
problem Optimizing multi-objective interval-valued functions on Hadamard manifolds.
method Developed KKT conditions for Pareto optimal solutions under different ordering and convexity notions.
result Results are more general than on Euclidean spaces.
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.
Study Hardy identities and inequalities on Cartan-Hadamard manifolds.
problem Existence and nonexistence of extremal functions in Hardy inequalities.
method Using the notion of a Bessel pair, we derive Hardy identities and inequalities.
result Established several Hardy type inequalities with improvements and understandings.
Cartan-Hadamard manifold is a simply connected Riemannian manifold with non-positive sectional curvature. In this article, we have proved that a Cartan-Hadamard manifold satisfying steady gradient Ricci soliton with the integral condition of potential function is isometric to the Euclidean space. Next we have proved a …
New theorem extends Hadamard's to transversely affine geometry.
problem Transversely affine foliations and their properties.
method Introduced and investigated novel transversely affine foliations, extending Hadamard's theorem.
result Transversely affine version of Hadamard's theorem for foliations.
Answering a question by Margulis we prove that the conclusion of Selberg's Lemma fails for discrete isometry groups of negatively curved Hadamard manifolds.
Proves Hadamard states for Dirac fields on manifolds with timelike boundaries.
problem Existence of Hadamard states for Dirac fields with MIT boundary conditions.
method Introducing geometric Møller operator to implement unitary isomorphism between spaces of initial data.
result Existence of Hadamard states for Dirac fields with MIT boundary conditions.
Using the classical approach we show the existence of disc type solutions to the asymptotic Plateau problem in certain Hadamard manifolds which may have arbitrarily strong curvature and volume growth.
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.
The paper provides formulas for Hadamard coefficients using Green's operators.
problem Calculating Hadamard coefficients from Green's operators.
method Various methods including resolvents, powers of Green's operators, and product with the real line.
result Formulas for Hadamard coefficients in terms of Green's operators.
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.
Study fourth order Schrödinger equation on Cartan-Hadamard manifolds, proving existence, scattering, and blow-up results.
problem Fourth order Schrödinger equation with mixed dispersion on Cartan-Hadamard manifolds.
method Fourier transform for hyperbolic space, weighted Strichartz estimates for rotationally symmetric manifolds, localized virial argument.
result Existence, scattering, and blow-up results for the equation.
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.
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.
We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold for a large class of operators containing in particular the p-Laplacian and the minimal graph operator.
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.
In this paper, we study harmonic functions on weighted manifolds and harmonic maps from weighted manifolds into Hadamard spaces introduced by Korevaar and Schoen. We prove Liouville theorems for these harmonic maps with finite energy.
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.
Locally convex compact immersed hypersurfaces in Finsler-Hadamard manifolds with bounded T-curvature are considered. We prove that such hypersurfaces are embedded as the boundary of convex body under certain conditions on the normal curvatures
Study shows curvature bounds for convex hypersurfaces in specific manifolds.
problem Bounding total curvature of convex hypersurfaces in Cartan-Hadamard manifolds.
method Analyzes curvature properties and applies Borbély's theorem.
result Total curvature is bounded below by the volume of the unit sphere.
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…
In this paper, we investigate submanifolds with locally bounded mean curvature in Hadamard manifolds, product manifolds N×R, submanifolds with bounded φ-mean curvature in the hyperbolic space, and successfully give lower bounds for the weighted fundamental tone and the first eigenvalue of the $p…
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.
The paper classifies Cartan-Hadamard manifolds supporting optimal Sobolev inequalities.
problem Classifying Cartan-Hadamard manifolds supporting optimal Sobolev inequalities.
method Analyzing the critical p-Laplace equation and its radial solutions.
result The only Cartan-Hadamard manifold supporting an optimal function for the Sobolev inequality is \( \mathbb{R}^n \).
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.
KReTTaH uses tensor trains and Hadamard overparameterization for fast, interpretable multi-way data imputation.
problem Multi-way data imputation for high-dimensional functional MRI and dynamic graph recovery.
method Reformulates imputation as RKHS regression with TT-constrained coefficients and Hadamard overparameterization. Optimizes TT coefficients and kernel matrices on Riemannian manifolds.
result Consistently outperforms state-of-the-art methods in modeling accuracy.
KReTTaH uses tensor trains and Hadamard overparameterization for fast, interpretable multi-way data imputation.
problem Multi-way data imputation in high-dimensional spaces.
method Reformulates imputation as RKHS regression with TT-constrained coefficients, optimized on manifold frameworks.
result Consistently outperforms state-of-the-art methods in accuracy.
Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.
problem Estimating spectral gaps for higher-order operators on Cartan-Hadamard manifolds.
method Symmetrization-free proofs based on general functional inequalities.
result Solves a sharp asymptotic problem from Cheng and Yang and answers a question from Kristály.
We proof existence theorems for the Dirichlet problem for hypersurfaces of constant special Lagrangian curvature in Hadamard manifolds. The first results are obtained using the continuity method and approximation and then refined using two iterations of the Perron method. The a-priori estimates used in the continuity m…
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. We show, by modifying Borbély's example, that there are 3-dimen\-sional Cartan-Hadamard manifolds M, with sectional curvatures ≤−1, such that the asymptotic Dirichlet problem for a class of quasilinear elliptic PDEs, including the minimal graph equation, is not solvable.
We study the filling invariants at infinity div_{k} for Hadamard manifolds defined by Brady and Farb in [Filling-invariants at infinity for manifolds of nonpositive curvature]. Among other results, we give a positive answer to the question they posed: whether these invariants can be used to detect the rank of a symmetr…
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
We present a new method for manufacturing complex-valued harmonic morphisms from a wide class of Riemannian Lie groups. This yields new solutions from an important family of homogeneous Hadamard manifolds. We also give a new method for constructing left-invariant foliations on a large class of Lie groups producing harm…