arXiv research

Paper uses ABP method to prove logarithmic Sobolev inequalities on curved spaces.

Logarithmic Sobolev inequality proven for non-compact self-shrinkers.

Through the main example of the Ornstein-Uhlenbeck semigroup, the Bakry-Emery criterion is presented as a main tool to get functional inequalities as Poincaré or logarithmic Sobolev inequalities. Moreover an alternative method using the optimal mass transportation, is also given to obtain the logarithmic Sobolev inequa…

Proves inequality for submanifolds with constant mean curvature.

We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.

Sharp inequality for submanifolds in curved spaces.

In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves…

Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.

In this paper we present our results on the logarithmic Sobolev inequality along the Ricci flow in dimension 2.

Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.

Modified Bakry-Émery criterion inequality for Tsallis entropy monotonicity.

We extend our previous results on the logarithmic Sobolev inequality along the Ricci flow in the case $λ_0(g_0)>0$ to the case $λ_0(g_0)=0$.

In this paper, we prove the concavity of $p$-entropy power of probability densities solving the $p$-heat equation on closed Riemannian manifold with nonnegative Ricci curvature. As applications, we give new proofs of $L^p$-Euclidean Nash inequality and $L^p$-Euclidean Logarithmic Sobolev inequality, moreover, an improv…

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

Researchers prove rigidity for log-Sobolev inequality on specific metric spaces.

In this survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates along the Ricci flow, including the Ricci flow with surgeries.

Ancient Ricci flows with asymptotic solitons have uniform bounds and inequalities.

Study bounds for Brownian motion on manifolds with sticky boundary conditions.

The paper proves various inequalities on gradient shrinking Ricci solitons.

This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine $L^{p}-$Sobolev inequalities. The logarithmic version of affine $L^{p}-$Sobolev inequalities is verified. Moreover, An alternative proof of the affine Mo…

We derive a logarithmic Sobolev inequality along the Ricci flow without any restriction on time, which depends only on the initial metric via rudimentary geometric data, assuming only that a certain first eigenvalue is positive. As a consequence we obtain a uniform Sobolev inequality along the Ricci flow without any re…

Sharp $L^p$-logarithmic-Sobolev inequalities on submanifolds with applications to hypercontractivity.

Study inequalities on hyperbolic spaces and Riemannian manifolds using symmetrization and heat semigroup.

We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying $\mathrm{Ric}_{\infty} \ge K>0$. Assuming equality holds, we show that the $1$-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng--Zhou on the spectral gap …

Ancient Ricci flows with bounded Nash entropy have uniform Sobolev inequalities.

Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,θ)$, we study the existence of a contact structure conformal to $θ$ for which the logarithmic Hardy-Littlewood-Sobolev (LHLS) inequality holds. Our approach closely follows \cite{Ok1} in the Riemannian setting. For this purpose, we introduce the notion …

Sharp upper diameter limit found for Ricci solitons.

For some class of geometric flows, we obtain the (logarithmic) Sobolev inequalities and their equivalence up to different factors directly and also obtain the long time non-collapsing and non-inflated properties, which generalize the results in the case of Ricci flow or List-Ricci flow or harmonic-Ricci flow. As applic…

The paper proves local rigidity theorems for scalar curvature and related inequalities.

This paper explains a technique for proving geometric inequalities.

Sharp Sobolev inequalities proved on manifolds with non-negative Ricci curvature.

The paper proves LOO CV is reliable under estimator stability.

New framework improves EM algorithm convergence under log-Sobolev inequality.

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…

A new algorithm solves semidefinite programs using Langevin diffusion.

The paper extends entropy formulas to super Ricci flows on metric measure spaces.

Method identifies low-dimensional structure in high-dimensional probability measures.

The ABP method is used to prove geometric inequalities for submanifolds and tensors.

The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.

An affine rearrangement inequality is established which strengthens and implies the recently obtained affine Pólya--Szegö symmetrization principle for functions on $\mathbb{R}^n$. Several applications of this new inequality are derived. In particular, a sharp affine logarithmic Sobolev inequality is established which i…

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

The paper proves curvature rigidity for manifolds with specific scalar curvature bounds.

Rapid mixing of Langevin dynamics on Riemannian manifolds

We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with low…

Study of diffusion annealed Langevin dynamics for generative models.

Greedy MI maximization method outperforms existing approaches in nonlinear models.

Proves upper bounds for heat kernels evolving on manifolds.

Let M be a compact n-dimensional manifold, $n\ge 2$, with metric g(t) evolving by the Ricci flow $\partial g_{ij}/\partial t=-2R_{ij}$ in (0,T) for some $T\in\Bbb{R}^+\cup\{\infty\}$ with $g(0)=g_0$. Let $λ_0(g_0)$ be the first eigenvalue of the operator $-Δ_{g_0} +\frac{R(g_0)}{4}$ with respect to g_0. We extend a rec…