arXiv research

The study improves Bochner inequality on Finsler manifolds to derive important inequalities.

The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.

We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the $Γ$-calculus a la Bakry et al, we show the dimensional versions of the Poincare--Lichnerowicz inequality, the logarithmic Sobolev…

It is known that by dualizing the Bochner-Lichnerowicz-Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalize…

Given a probability measure $μ$ supported on a convex subset $Ω$ of Euclidean space $(\mathbb{R}^d,g_0)$, we are interested in obtaining Poincaré and log-Sobolev type inequalities on $(Ω,g_0,μ)$. To this end, we change the metric $g_0$ to a more general Riemannian one $g$, adapted in a certain sense to $μ$, and perform…

Paper proves rigidity of metrics near hyperbolic ones in 3D.

We prove regularity for a class of boundary value problems for first order elliptic systems, with boundary conditions determined by spectral decompositions, under coefficient differentiability conditions weaker than previously known. We establish Fredholm properties for Dirac-type equations with these boundary conditio…

Introduces a universal Bochner formula for scalar curvature.

Uniform Poincaré inequalities established for various metric spaces.

The study improves Poincaré and log-Sobolev inequalities on hyperbolic spaces.

New rigidity results for tensors on non-compact manifolds with curvature conditions.

New proof shows inequality without restrictions.

We prove a Poincare type inequality for differential forms on compact manifolds by means of a constructive 'globalization' of a local Poincare inequality on convex sets.

Study Poincaré inequality in metric spaces via separating sets.

We investigate the possibility of improving the $p$-Poincaré inequality $\|\nabla_{\mathbb{H}^N} u\|_p \ge Λ_p \|u\|_p$ on the hyperbolic space, where $p>2$ and $Λ_p:=[(N-1)/p]^{p}$ is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is …

Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincaré inequality", we provide a simple example of a metric space $X$ that admits Poincaré inequalities for a continuum of mutually singular measures.

New Poincaré inequality for differential forms on manifolds.

Proves inequality linking function deviation to gradient norm on compact manifolds.

The paper proves new inequalities for convex hypersurfaces using centro-affine geometry.

The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.

We construct Dirac operators on foliations by applying the Bismut-Lebeau analytic localization technique to the Connes fibration over a foliation. The Laplacian of the resulting Dirac operators has better lower bound than that obtained by using the usual adiabatic limit arguments on the original foliation. As a consequ…

We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, $n$-dimensional Riemannian manifold with pinched negative sectional curvature follows …

Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.

We study the validity of the $L^p$ inequality for the Riesz transform when $p>2$ and of its reverse inequality when $p<2$ on complete Riemannian manifolds under the doubling property and some Poincaré inequalities.

The article proves a Poincaré inequality for hypersurfaces and applies it to rigidity results.

The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.

Study complete manifolds with weighted Poincaré inequality and Ricci curvature bounds.

The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.

We show that, on any asymptotically hyperbolic surface, the essential spectrum of the Lichnerowicz Laplacian $Δ_L$ contains the ray $[{1/4},+\infty[$. If moreover the scalar curvature is constant then -2 and 0 are infinite dimensional eigenvalues. If, in addition, the inequality $<Δu, u>_{L^2}\geq \frac14||u||^2_{L^2}$…

The development of global sensitivity analysis of numerical model outputs has recently raised new issues on 1-dimensional Poincaré inequalities. Typically two kind of sensitivity indices are linked by a Poincaré type inequality, which provide upper bounds of the most interpretable index by using the other one, cheaper …

A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poinca…

Short note proves Poincaré inequality for 4-manifold forms.

Study on functional inequalities on simple edge spaces.

We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincaré inequality. In the process, a sharp decay estimate for the minimal positive Green's function is obtained. This estimate only depends on the weight function of the Poincaré inequality, and yields …

New proof of surface group theorem for 2D Poincaré duality groups.

We show that any $d$-Ahlfors regular subset of $\mathbb{R}^{n}$ supporting a weak $(1,d)$-Poincaré inequality with respect to surface measure is uniformly rectifiable.

Sharp inequality for compactifying Poincaré-Einstein manifolds.

This paper extends the convergence analysis of Langevin Monte Carlo beyond Poincaré inequalities.

Sharp lower bound found for geodesic ball eigenvalues.

The study proves topological rigidity for certain geometric shapes using Poincaré inequalities.

The paper derives inequalities and formulas for generalized Ricci flow.

Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.

In this paper we prove $L^{\infty}$ type gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincaré inequality. We apply the theorems to a broad class of complete manifolds satisfying weighted Poincaré inequalities. In particular, we obtain a gap theorem on the Euclidean space wi…

The paper proves optimizability implies inequalities for sampling.

We study some equivalent properties of the curvature-dimension conditions $CD(n,K)$ inequality on infinite, but locally finite graph. These equivalences are gradient estimate, Poincaré type inequalities and reverse Poincaré inequalities. And we also obtain one equivalent property of gradient estimate for a new notion o…

We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.

In this paper, we prove Poincaré and Sobolev inequalities for differential forms in $L^1(\mathbb R^n)$. The singular integral estimates that it is possible to use for $L^p$, $p>1$, are replaced here with inequalities which go back to Bourgain-Brezis.

In this paper, we prove several Poincaré inequalities of fractional type on conformally flat manifolds with finite total Q-curvature. This shows a new aspect of the $Q$-curvature on noncompact complete manifolds.