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.
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.
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.
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.
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…
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.
We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance to a fixed point in M. We are, in particular, interested in finding optimal (or…
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 article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold (M,g) corresponding to e…
Study Liouville theorem for specific harmonic maps.
problem Proving Liouville theorem for harmonic maps.
method Establish gradient estimates under specific conditions.
result Proved Liouville theorem for harmonic maps.
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.
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.
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.
If (Nm+p,h) is a Cartan-Hadamard manifold such that Ric(h)≥−G(rN(x)) where G(0)≥1,G′≥0 and G−1/2∈L1(+∞) then every proper biharmonic isometric immersion φ:Mm→(Nm+p,h) is a harmonic map.
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.
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.
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…
The paper defines a universal Teichmüller space for PGL_d(R) and proves its properties.
problem Defining a universal Teichmüller space for PGL_d(R).
method Using harmonic maps and stability criteria for coarse Lipschitz maps.
result Defines a universal Teichmüller space for PGL_d(R) and proves its properties.
We show that for every quasi-isometric map from a Hadamard manifold of pinched negative curvature to a locally compact, Gromov hyperbolic, CAT(0)-space there exists an energy minimizing harmonic map at finite distance. This harmonic map is moreover Lipschitz. This generalizes a recent result of Benoist-Hulin.
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 …
Sharp inequality proved in 3D hyperbolic spaces using flow methods.
problem Proving a sharp Minkowski-type inequality in Cartan-Hadamard 3-spaces.
method Using harmonic mean curvature flow to prove the inequality.
result Sharpened estimates for total mean curvature in hyperbolic 3-space.
Study spherical Fourier transform on hypergeometric type harmonic manifolds.
problem Spherical Fourier transform on harmonic Hadamard manifolds.
method Representation of spherical functions by Gauss hypergeometric functions.
result Inversion formula, convolution rule, and Plancherel theorem are derived.
We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold M with only one end if M has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constan…
New principle for harmonic maps helps study higher-dimensional submanifolds.
problem Understanding unboundedness of totally geodesic projections in higher codimension.
method Introducing a flexible notion of convexity and applying it to harmonic and conformal maps.
result New maximum principle for harmonic maps applicable to various geometric settings.
The Martin boundary of a Cartan-Hadamard manifold describes a fine geometric structure at infinity, which is a sub-space of positive harmonic functions. We describe conditions which ensure that some points of the sphere at infinity belong to the Martin boundary as well. In the case of the universal cover of a compact m…
We construct harmonic diffeomorphisms from the complex plane C onto any Hadamard surface M whose curvature is bounded above by a negative constant. For that, we prove a Jenkins-Serrin type theorem for minimal graphs in M×R over domains of M bounded by ideal geodesic polygons and show the existence of a se…
We show that the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data, for certain range of p>n, on an n-dimensional Cartan-Hadamard manifold constructed from a complete noncompact shrinking gradient Ricci soliton. Using the steady gradient Ricci soliton, w…
Study harmonic maps and anti-de Sitter 3-manifolds.
problem Existence and uniqueness of harmonic maps in infinite energy settings.
method Generalization of Corlette's result to infinite energy, analysis of asymptotic behavior, use of CAT(-1) Hadamard manifolds.
result Existence of new anti-de Sitter 3-manifolds.
It is proved the existence of entire solutions of the Laplace's and minimal hypersurface's PDEs on a Hadamard manifold M under certain curvature conditions by investigating the asymptotic Dirichlet's problems for these PDEs. In the harmonic case it is obtained an existence result which assumes the same growth conditi…
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.
We study the asymptotic Dirichlet problem for A-harmonic functions on a Cartan-Hadamard manifold whose radial sectional curvatures outside a compact set satisfy an upper bound K(P)≤−r(x)2logr(x)1+ε and a pointwise pinching condition ∣K(P)∣≤CK∣K(P′)∣ for some const…
Existence of harmonic maps near projections in hyperbolic spaces for large convex sets.
problem Existence of harmonic maps near projections in hyperbolic spaces.
method Weak quasi-isometry condition, non-collapsing property, harmonic measures.
result Existence of harmonic maps at finite distance from projections of large convex sets in hyperbolic spaces.
We consider a class of martingales on Cartan-Hadamard manifolds that includes Brownian motion on a minimal submanifold. We give sufficient conditions for such martingales to be transient, extending previous results on the transience of minimal submanifolds. We also give conditions for the almost sure convergence of the…
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.
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.
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.
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…
We study minimal energy problems for strongly singular Riesz kernels on a manifold. Based on the spatial energy of harmonic double layer potentials, we are motivated to formulate the natural regularization of such problems by switching to Hadamard's partie finie integral operator which defines a strongly elliptic pseud…
The article proves isometry theorems for specific types of manifolds.
problem Investigating properties of Cartan-Hadamard manifolds and related solitons.
method Analyzing steady, gradient shrinking, and expanding Ricci solitons.
result Specific manifolds are isometric to Euclidean space under certain conditions.
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 convex functions on manifolds without focal points, deriving new spectral properties.
problem Convex functions on manifolds without focal points.
method Geometrically defined convex functions, spectral analysis.
result Spectrum is purely absolutely continuous on certain manifolds.
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.