arXiv research

Proves inequality for submanifolds with constant mean curvature.

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

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

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

Modified Bakry-Émery criterion inequality for Tsallis entropy monotonicity.

Paper disproves conjecture about log-Sobolev constants.

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

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

New Dynkin condition for manifolds with boundary yields bi-Lipschitz equivalence and spectral properties.

In this paper we introduce the log entropy functional and establish its monotonicity along the Ricci flow. One consequence of it is the monotonicity of the logarithmic Sobolev constant along the Ricci flow.

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 …

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

In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estim…

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…

Sharp Sobolev inequalities proved on manifolds with non-negative Ricci 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 paper we present our results on the logarithmic Sobolev inequality along the Ricci flow in dimension 2.

Study of conformal logarithmic Laplacian on sphere, connecting Yamabe problems and Sobolev spaces.

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

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

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 $λ_0(g_0)=0$.

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

Upper bounds on constants for Brownian motion with sticky boundary.

This work shows linear convergence for two-layer neural networks in mean-field regime.

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

Intertwining curvature bounds for graphs and quantum Markov semigroups verified.

LMC algorithm converges to target in Chi-squared and Renyi divergence.

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…

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…

Smooth activations enable optimal error rates in neural networks for Sobolev function classes.

Improved particle approximation for mean-field neural networks.

Study nonexistence and gradient estimates for solutions on manifolds with bounded Ricci curvature.

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

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.

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…

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 …

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

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

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…

Deep ReLU networks can efficiently approximate Sobolev and Besov functions.

Sharp Lipschitz bounds for flow-matching and diffusion models with optimal sampling rates.

The paper proves various inequalities on gradient shrinking Ricci solitons.

A new algorithm solves semidefinite programs using Langevin diffusion.

We define the "sum of squares of the wavelengths" of a Riemannian surface (M,g) to be the regularized trace of the inverse of the Laplacian. We normalize by scaling and adding a constant, to obtain a "mass", which is scale invariant and vanishes at the round sphere. This is an anlaog for closed surfaces of the ADM mass…

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