arXiv research

Solves Plateau problem for surfaces in pinched curvature manifolds.

Harmonic maps between pinched Hadamard surfaces are quasi-conformal.

We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) 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, $\mathbb{C}$) 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…

Maps between Hadamard manifolds are quasi-isometric to harmonic maps.

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…

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.

One-harmonic maps from a curved surface to hyperbolic plane have specific interior properties.

We give upper and lower bounds for the ratio of the volume of metric ball to the area of the metric sphere in Finsler-Hadamard manifolds with pinched S-curvature. We apply these estimates to find the limit at the infinity for this ratio. Derived estimates are the generalization of the well-known result in Riemannian ge…

The paper establishes eigenvalue inequalities for a specific operator on curved spaces.

Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.

Constructs harmonic maps near retractions in hyperbolic spaces.

We show that for every quasi-isometric map from a Hadamard manifold of pinched negative curvature to a locally compact, Gromov hyperbolic, ${\rm CAT}(0)$-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.

Let $X$ be an $n$-dimensional simply connected manifold of pinched sectional curvature $-a^2 \leq K \leq -1$. 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 $Ent (Γ) \geq C(n,a)$.

Study on minimal submanifolds in curved spaces with unique solution to asymptotic Plateau problem.

In this paper, we prove a quantitative version of the Tits alternative for negatively pinched manifolds $X$. Precisely, we prove that a nonelementary discrete isometry subgroup of $\mathrm{Isom}(X)$ generated by two non-elliptic isometries $g$, $f$ contains a free subgroup of rank $2$ generated by isometries $f^N , h$ …

In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of Cartan-Hadamard, Synge, and Bonnet-Myers as well as a generalization of the (non-smooth)…

We prove mean curvature estimates and a Jorge-Koutroufiotis type theorem for submanifolds confined into either a horocylinder of N X L or a horoball of N, where N is a Cartan-Hadamard manifold with pinched curvature. Thus, these submanifolds behave in many respects like submanifolds immersed into compact balls and into…

We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan-Hadamard manifold $M$ whose radial sectional curvatures outside a compact set satisfy an upper bound $K(P)\le - \frac{φ(φ-1)}{r(x)^2}$ and a pointwise pinching condition $|K(P)|\le C_K|K(P')|$ for some constants $φ>1$ and $C_K\ge 1…

We study the asymptotic Dirichlet problem for $\mathcal{A}$-harmonic functions on a Cartan-Hadamard manifold whose radial sectional curvatures outside a compact set satisfy an upper bound $K(P)\le - \frac{1+\varepsilon}{r(x)^2 \log r(x)}$ and a pointwise pinching condition $|K(P)|\le C_K |K(P')|$ for some const…

In this paper we will prove Hadamard-Stoker type theorems in the following ambient spaces: $\man ^n \times \r$, where $\man ^n $ is a $1/4-$pinched manifold, and certain Killing submersions, e.g., Berger spheres and Heisenberg spaces. That is, under the condition that the principal curvatures of an immersed hypersurfac…

Let $Γ$ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold $X$. We show that a normal subgroup $Γ_0$ has critical exponent equal to the critical exponent of $Γ$ if and only if $Γ/ Γ_0$ is amenable. We prove a similar result for the exponential growth rate of closed geodesics on $…

Let $X$ be a Hadamard manifold, and $Γ$ a non-elementary discrete group of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/Γ$ to the behavior of the Poincar{é} series of $Γ$. Precisely, the aim of this paper is to extend the so-called…

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…

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 Poisson kernel on negatively curved Hadamard manifolds.

For a pinched Hadamard manifold $X$ and a discrete group of isometries $Γ$ of $X$, the critical exponent $δ_Γ$ is the exponential growth rate of the orbit of a point in $X$ under the action of $Γ$. We show that the critical exponent for any family $\mathcal{N}$ of normal subgroups of $Γ_0$ has the same coarse behaviour…

Estimates gaps between eigenvalues for elliptic operators on manifolds.

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 $K \le κ\le 0$, 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…

Study shows spectrum properties for specific Hadamard manifolds.

Sharp curvature inequality extends Euclidean isoperimetric inequality to Cartan-Hadamard manifolds.

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…

The study examines the normal growth exponent of submanifolds in negatively curved manifolds.

Uniformly extend maps on Hadamard manifolds with curvature constraints.

Study pinched self-dual Weyl curvature in compact 4-manifolds.

Three-manifolds with non-negative pinched Ricci curvature have complete Ricci flows.

Flat Yang-Mills connections on pinched 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…

Existence of harmonic maps near projections in hyperbolic spaces for large convex sets.

The paper proves Liouville-type theorems on Hadamard manifolds.

We refine a metric bunching estimate for pinched manifolds.

This paper extends 3D results to higher dimensions, proving compactness for PIC1 pinched manifolds.

The article proves isometry theorems for specific types of manifolds.

Alternative proof of flatness for Ricci-pinched 3-manifolds.

Upper bounds for circumradius in Hadamard surfaces with curvature constraints.

The study extends isoperimetric inequalities to non-positive curvature spaces.

The paper defines Fenchel conjugate and biconjugate on Hadamard manifolds.

Nonexistence of radial optimal functions on certain Cartan-Hadamard manifolds.