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.
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.
We provide elementary proofs of Lemmas 7.1 and 7.4 appearing in "The Cartan-Hadamard conjecture and the Little Prince", by B. Kloeckner and G. Kuperberg. The Lemmas play an important role in the derivation of novel isoperimetric inequalities. The original proofs relied on Sage, a symbolic algebra package, to factor cer…
Local isoperimetric inequality holds for balls with nonpositive curvature.
problem Preserving the isoperimetric ratio in perturbed ball metrics with nonpositive curvature.
method Analyzing perturbations of ball metrics with nonpositive curvature.
result Isoperimetric ratio is preserved only by homotheties of the ball.
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 \( \mathbb{R}^n \).
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…
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.
The paper establishes eigenvalue inequalities for a specific operator on curved spaces.
problem Eigenvalue estimation for a specific operator on curved domains.
method Bochner type formula and Rauch comparison theorem.
result Universal inequalities for eigenvalues of the drifted Cheng-Yau operator.
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.
We study Lord Rayleigh's problem for clamped plates on an arbitrary n-dimensional (n≥2) Cartan-Hadamard manifold (M,g) with sectional curvature K≤−κ2 for some κ≥0. We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in (M,g) is universally bounde…
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.
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.
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.
Sharp isoperimetric inequality derived from Q-curvature for smooth metrics.
problem Deriving a sharp isoperimetric inequality from the top order Q-curvature.
method Analyzing the relationship between Q-curvature and sectional curvature, applying isoperimetric inequality.
result Sharp isoperimetric inequalities derived for domains with smooth boundaries.
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.
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.
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.
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.
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.
Total curvatures of certain hypersurfaces are continuous.
problem Continuity of curvatures in geometric settings.
method Hausdorff distance for hypersurfaces and convex bodies in Riemannian manifolds and Cartan-Hadamard spaces.
result Total generalized mean curvatures are continuous.
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.
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.
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.
Proves a synthetic Lorentzian Cartan-Hadamard theorem.
problem Formulates and proves a theorem for Lorentzian geometry.
method Uses an appropriate notion of local concavity for Lorentzian (pre-)length spaces.
result Establishes existence and uniqueness of timelike geodesics.
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.
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…
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
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.
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.
Study counts surface subgroups in curved 3D manifolds.
problem Count surface subgroups in curved 3D manifolds.
method Solve foliated Plateau problem in Cartan-Hadamard manifolds.
result Prove rigidity for lower bound on surface subgroup count.
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…
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.
Study investigates non-existence of bounded solutions on curved spaces.
problem Non-existence of bounded solutions to semi-linear elliptic equations on Cartan-Hadamard manifolds.
method Novel comparison technique using convex hypersurfaces.
result Extends previous results to curved spaces, highlighting curvature's role.
Study finds conditions for global minimizers on curved manifolds with fast diffusion and nonlocal interactions.
problem Existence of global minimizers for a free energy functional on negatively curved manifolds.
method Investigation of Carlson-Levin type inequalities for Cartan-Hadamard manifolds.
result Establishes necessary and sufficient conditions for the existence of global energy minimizers.
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…
We give a general criterion for the Dirichlet problem at infinity (DPI) on a Cartan-Hadamard surface to be solvable, which we primarily use to give the best possible upper radial radial curvature bound for solvability of the DPI, but which is also flexible enough to accommodate flats. In particular, any (upper) radial …
The paper derives inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.
problem Geometric inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.
method Comparison formula via Reilly's identities; geometric inequalities derived.
result Sharp lower bound for total first mean curvature in dimension 3.
New Sliced-Wasserstein distances for non-Euclidean data.
problem Computational burden of Wasserstein distance on non-Euclidean manifolds.
method Derive Sliced-Wasserstein distances and flows on Cartan-Hadamard manifolds.
result General constructions and non-parametric schemes for minimizing new distances.
Lower bounds on average normal curvature for submanifolds in Riemannian domains.
problem Finding bounds on the average normal curvature of submanifolds in Riemannian domains.
method Using an invariant measuring optimal n-trace convexity under unit-gradient normalization. result Lower bounds for the average normal curvature expressed in terms of an invariant.
Paper proves uniqueness of minimal maps in curved spaces.
problem Proving uniqueness of minimal maps into Cartan-Hadamard manifolds.
method Proof based on convexity of functions in terms of squared singular values.
result Uniqueness theorem for minimal maps into Riemannian manifolds.
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…
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.
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…
The paper explores how Finsler manifolds differ from Riemannian ones in functional inequalities.
problem Analytic phenomena on Finsler manifolds differ from Riemannian ones.
method Comparative analysis of Finsler and Riemannian manifolds.
result Functional inequalities (Hardy, uncertainty, CKN) behave differently on Finsler manifolds.