arXiv research

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

Solves Plateau problem for surfaces in pinched curvature manifolds.

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

Closed surfaces minimize total curvature in curved spaces.

Sharp Minkowski inequality for convex surfaces in curved spaces.

Study Hardy identities and inequalities on Cartan-Hadamard manifolds.

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 entropy functionals for curved spaces help predict shape behavior.

Study finds solitons on curved spaces with varying behavior.

Study fourth order Schrödinger equation on Cartan-Hadamard manifolds, proving existence, scattering, and blow-up results.

Convex hypersurfaces in curved spaces bound convex regions.

Study on existence of ground states on curved spaces with conditions on potential growth.

Study shows curvature bounds for convex hypersurfaces in specific manifolds.

It is well known that the Euclidean Sobolev inequality holds on any Cartan-Hadamard manifold of dimension $n\ge 3$, i.e. any complete, simply connected Riemannian manifold with nonpositive sectional curvature. As a byproduct of the Cartan-Hadamard conjecture, a longstanding problem in the mathematical literature sett…

Estimates prove existence of curvature flow in curved spaces.

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…

The paper classifies Cartan-Hadamard manifolds supporting optimal Sobolev inequalities.

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.

Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.

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 $K_{N}\leq b<0$

If $(N^{m+p},h)$ is a Cartan-Hadamard manifold such that $Ric(h)\geq -G(r_{N}(x))$ where $G(0)\geq 1, G^{'}\geq 0$ and $G^{-1/2}\not\in L^{1}(+\infty)$ then every proper biharmonic isometric immersion $φ: M^{m}\rightarrow(N^{m+p},h)$ is a harmonic map.

Study on radial solutions of Lane-Emden system on Cartan-Hadamard manifolds.

The paper examines functional properties on manifolds with very negative curvature.

Study finds conditions for global minimizers on curved manifolds with fast diffusion and nonlocal interactions.

Total curvatures of certain hypersurfaces are continuous.

After recalling the Dirichlet problem at infinity on a Cartan-Hadamard manifold, we discuss what is known and the difference between the two-dimensional and higher-dimensional cases. Turning our attention to the two-dimensional case, we prove that the Dirichlet problem at infinity on a two-dimensional Cartan-Hadamard m…

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…

Study investigates non-existence of bounded solutions on curved spaces.

We study the geodesic X-ray transform on Cartan-Hadamard manifolds, and prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature de…

Finite time for subsolutions on Riemannian manifolds proved.

The paper derives inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.

New Sliced-Wasserstein distances for non-Euclidean data.

The paper explores how Finsler manifolds differ from Riemannian ones in functional inequalities.

The paper explores how Finsler manifolds differ from Riemannian ones in functional inequalities.

Paper proves uniqueness of minimal maps in curved spaces.

We give a survey on the development of the study of the asymptotic Dirichlet problem for the minimal surface equation on Cartan-Hadamard manifolds. Part of this survey is based on the introductory part of the doctoral dissertation of the author. The paper is organised as follows. First we introduce Cartan-Hadamard mani…

The aim of this paper is to obtain the fundamental tone for minimal submanifolds of the Euclidean or hyperbolic space under certain restrictions on the extrinsic curvature. We show some sufficient conditions on the norm of the second fundamental form that allow us to obtain the same upper and lower bound for the fundam…

Lower bounds on average normal curvature for submanifolds in Riemannian domains.

We explore the consequences of curvature and torsion on the topology of quaternionic contact manifolds with integrable vertical distribution. We prove a general Myers theorem and establish a Cartan-Hadamard result for almost qc-Einstein manifolds.

In this paper, we prove a version of the classical Cartan-Hadamard theorem for negatively curved manifolds, of dimension $n\neq 5$, with non-empty totally geodesic boundary. More precisely, if $M_1^n,M_2^n$ are any two such manifolds, we show that (1) $\partial ^\infty \tilde M_1^n$ is homeomorphic to $\partial ^\infty…

3-manifolds with convex boundary are rigid in certain curvature conditions.

Sharp inequality proved in 3D hyperbolic spaces using flow methods.

We obtain an explicit formula for comparing total curvature of level sets of functions on Riemannian manifolds, and develop some applications of this result to the isoperimetric problem in spaces of nonpositive curvature.

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.

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…

We provide uniqueness results for compact minimal submanifolds in a large class of Riemannian manifolds of arbitrary dimension. In the case compact and Cartan-Hadamard manifolds we obtain general results for these submanifolds. Several applications to Geometric Analysis are also showed.

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

New inequalities for submanifolds in curved spaces.