Formulae for divergent integrals with singularities, useful in string theory.
problem Handling divergent integrals with singularities.
method Formulae for finite part of divergent integrals, using local residue map.
result Formulae for finite part of divergent integrals expressed in terms of local residue map.
Survey on minimal surface equation on Cartan-Hadamard manifolds.
problem Asymptotic Dirichlet problem for minimal surface equation.
method Introduction of Cartan-Hadamard manifolds, discussion of results and methods used in proofs.
result Discussion of results and methods used in proofs.
Formula for Hadamard coefficients from Green's operators on spacetimes.
problem Calculating Hadamard coefficients from Green's operators on spacetimes.
method Developed formulas for diagonal values and integrals over the diagonal of Hadamard coefficients.
result Formulated analogues of Hadamard expansions and resolvents for Green's operators.
The paper extends geometric finiteness to discrete subgroups of negatively pinched Hadamard manifolds.
problem Characterizing geometrically infinite discrete subgroups of negatively pinched Hadamard manifolds.
method Generalizing Bonahon's characterization and proving a theorem of Bishop's extension.
result Every discrete geometrically infinite isometry subgroup has a set of nonconical limit points of cardinality continuum.
Proves homological inequality for cycles in Hadamard spaces of asymptotic rank 2.
problem Establishing isoperimetric inequalities in Hadamard spaces of asymptotic rank two.
method Homological inequality for cycles in dimensions at least 2, assuming finite linearly controlled asymptotic dimension.
result Homological inequality for general cycles in Hadamard 3-manifolds and finite-dimensional CAT(0) cube complexes.
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 \( \mathbb{R}^n \).
Given a closed submanifold, or a compact regular domain, in euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two different regularization techniques (Hadamard's finite part and analytic continuatio…
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.
We use the combinatorial harmonic map theory to study the isometric actions of discrete groups on Hadamard spaces. Given a finitely generated group acting by automorphisms, properly discontinuously and cofinitely on a simplicial complex and its isometric action on a Hadamard space, we formulate criterions for the actio…
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 minimal surfaces with finite curvature in a specific manifold.
problem Understanding minimal surfaces with finite total curvature in a Hadamard manifold.
method Developed a formula to compute total curvature using topological, geometrical, and conformal data.
result Total curvature is an integral multiple of \(2\pi\).
Finite time for subsolutions on Riemannian manifolds proved.
problem Finite extinction time for subsolutions of a specific equation on Riemannian manifolds.
method Proved finite extinction time using weighted Sobolev inequality and assumptions on p, q, and ρ.
result Weak subsolutions to the equation have a finite extinction time.
We give an analytical proof of the Poincare-type inequalities for widths of geodesic homotopies between equivariant maps valued in Hadamard metric spaces. As an application we obtain a linear bound for the length of an element conjugating two finite lists in a group acting on an Hadamard space.
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.
Harmonic maps from curved spaces to hyperbolic ones exist and are Lipschitz.
problem Existence and properties of harmonic maps between curved and hyperbolic spaces.
method Energy minimization and finite distance analysis.
result Harmonic maps from Hadamard manifolds to Gromov hyperbolic mCAT(0)-spaces are Lipschitz. We show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN≤κ≤0 are proper, (compact if N is compact). In addition, if N is Hadamard then M has finite topology. We also show that the fundamental tone is an obstruction fo…
The purpose of this erratum is to correct the proof of Theorem A.0.1 in the appendix to our article ``Hadamard spaces with isolated flats'' math.GR/0411232, which was jointly authored by Mohamad Hindawi, Hruska and Kleiner. In that appendix, many of the results of math.GR/0411232 about CAT(0) spaces with isolated flats…
Study on radial solutions of Lane-Emden system on Cartan-Hadamard manifolds.
problem Existence and qualitative properties of radial solutions on Cartan-Hadamard manifolds.
method Analytical and asymptotic analysis of radial solutions, focusing on critical and supercritical exponents.
result Existence of one-parameter family of radial solutions for critical or supercritical exponents, with different dimensions of existence regions based on stochastic completeness.
New method for optimization on Hadamard manifolds with curvature-independent guarantees.
problem Curvature-dependent complexity in geodesic convex optimization.
method Introducing horospherical convexity and developing algorithms for optimization.
result Curvature-independent convergence of subgradient descent and Nesterov's method.
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.
Study shows critical exponents of normal subgroups behave similarly to Kazhdan distances.
problem Analyzing critical exponents and Kazhdan distances in pinched Hadamard manifolds.
method Analyzing the spectrum of transfer operators associated with subshifts of finite type.
result Critical exponents of normal subgroups have the same coarse behavior as Kazhdan distances.
Let Mm, with m≥3, be an m-dimensional complete noncompact manifold isometrically immersed in a Hadamard manifold Mˉ. Assume that the mean curvature vector has finite Lp-norm, for some 2≤p≤m. We prove that each end of M must either have finite volume or be non-parabolic.
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.
The paper derives expansions for Green's operators and resolvents using Hadamard methods.
problem Analyzing normally hyperbolic operators and their Green's functions.
method Hadamard expansions for powers of Green's operators and resolvents.
result Derives expansions involving Hadamard coefficients for advanced/retarded Green's operators.
Harmonic maps between pinched Hadamard surfaces are quasi-conformal.
problem Characterizing harmonic maps between Hadamard surfaces.
method Proving harmonic quasi-isometries are quasi-conformal diffeomorphisms.
result Harmonic quasi-isometries of pinched Hadamard surfaces are injective.
New insights into identifying mixtures of product distributions using Hadamard extensions.
problem Identifying mixtures of product distributions on binary variables.
method Analysis of Hadamard extensions of matrix products.
result Conditions for full column rank of Hadamard extensions.
Cyclic projections in Hadamard spaces can be irregular, unlike in Hilbert spaces.
problem Understanding the behavior of cyclic projections in Hadamard spaces compared to Hilbert spaces.
method Constructing an example of convex subsets in a Hadamard space.
result Cyclic product of projections is not asymptotically regular in Hadamard spaces.
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.
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.
The paper estimates heights of H-surfaces in warped products with conditions.
problem Estimating heights of constant mean curvature surfaces in warped products.
method Analyzing surfaces with transversal boundaries in warped product spaces.
result Height estimates involving area and bounded volume for H-surfaces.
We study the asymptotic Dirichlet problem for f-minimal graphs in Cartan-Hadamard manifolds M. f-minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. In the first part of this paper, we prove the existence of f-minimal graphs with pre…
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.
The generalized Cartan-Hadamard conjecture says that if Ω is a domain with fixed volume in a complete, simply connected Riemannian n-manifold M with sectional curvature K≤κ≤0, then the boundary of Ω has the least possible boundary volume when Ω is a round n-ball with constant curvature K=κ. The c…
Uniformly extend maps on Hadamard manifolds with curvature constraints.
problem Extending maps on Hadamard manifolds with curvature constraints.
method Proving uniform extension for contracting maps on Hadamard manifolds with curvature bounds.
result Uniform Lipschitz extension achieved for contracting maps on Hadamard manifolds with curvature constraints.
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.
We study the asymptotic Dirichlet and Plateau problems on Cartan-Hadamard manifolds satisfying the so-called Strict Convexity (abbr. SC) condition. The main part of the paper consists in studying the SC condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on…
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.
The paper proves Hardy and Rellich inequalities for submanifolds in Hadamard spaces.
problem Integral inequalities for submanifolds in Hadamard spaces.
method Proving Hardy and Rellich inequalities for submanifolds in Hadamard spaces.
result General Hardy and Rellich inequalities for submanifolds in Hadamard spaces.
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 studies submanifolds in Hadamard and product manifolds, finding eigenvalue bounds.
problem Eigenvalue estimates for submanifolds in specific geometric settings.
method Investigation of submanifolds with bounded mean curvature in Hadamard manifolds and product manifolds NimesR. result Lower bounds for the weighted fundamental tone and the first eigenvalue of the p-Laplacian. Paper computes link determinants using Fourier-Hadamard transforms.
problem Computing determinants of complex link structures.
method Fourier-Hadamard transforms of Boolean functions.
result Determinant of centrally symmetric links with even components equals zero.
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, …
Let X be an n-dimensional simply connected manifold of pinched sectional curvature −a2≤K≤−1. There exist a positive constant C(n,a) such that for any finitely generated discrete group Γ acting on X, then either Γ is virtually nilpotent or the algebraic entropy Ent(Γ)≥C(n,a).
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.
Paper proposes Walsh-Hadamard Variational Inference for efficient approximate inference in large models.
problem Over-regularization in variational inference for large models.
method Walsh-Hadamard factorization strategies to reduce parameterization, accelerate computations, and increase posterior expressiveness.
result Efficient approximate inference achieved in over-parameterized models.
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.