Related papers: Geometric inequalities and rigidity of gradient sh…

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,…

In this paper, we prove the equivalent of ultracontractive bound of heat semigroup or the uniform upper bound of the heat kernel with the Nash inequality, Log-Sobolev inequalities on graphs. We also show that under the assumption of volume…

Haslhofer and M\"uller proved a compactness Theorem for four-dimensional shrinking gradient Ricci solitons, with the only assumption being that the entropy is uniformly bounded from below. However, the limit in their result could possibly…

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work of Lopez-Rio imply rigidity of gradient shrinking Ricci solitons with harmonic Weyl…

We establish a uniform Sobolev inequality for K\"ahler metrics, which only require an entropy bound and no lower bound on the Ricci curvature. We further extend our Sobolev inequality to singular K\"ahler metrics on K\"ahler spaces with…

We prove a splitting theorem for complete gradient Ricci soliton with nonnegative curvature and establish a rigidity theorem for codimension one complete shrinking gradient Ricci soliton in $\mathbb R^{n+1}$ with nonnegative Ricci…

We prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in $\mathbb{R}^n$. In addtion, we partially give analogous rigidity results of the…

We prove that a $n$-dimensional, $4 \leq n \leq 6$, compact gradient shrinking Ricci soliton satisfying a $L^{n/2}$-pinching condition is isometric to a quotient of the round $\mathbb{S}^{n}$. The proof relies mainly on sharp algebraic…

We prove that for a symmetric Markov semigroup, Ricci curvature bounded from below by a non-positive constant combined with a finite $L_\infty$-mixing time implies the modified log-Sobolev inequality. Such $L_\infty$-mixing time estimates…

Suppose $(M^n, g, f)$ is a complete shrinking gradient Ricci soliton. We give several rigidity results under some natural conditions, generalizing the results in \cite{Petersen-Wylie,Guan-Lu-Xu}. Using maximum principle, we prove that…

In this work, we study the rigidity problem for the logarithmic Sobolev inequality on a complete metric measure space $(M^n,g,f)$ with Bakry-\'Emery Ricci curvature satisfying $Ric_f\geq \frac{a}{2}g$, for some $a>0$. We prove that if…

We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman's entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp…

We study integral and pointwise bounds on the curvature of gradient shrinking Ricci solitons. As applications we discuss gap and compactness results for gradient shrinkers.

We present a Gagliardo-Nirenberg inequality which bounds Lorentz norms of the function by Sobolev norms and homogeneous Besov quasinorms with negative smoothness. We prove also other versions involving Besov or Triebel-Lizorkin quasinorms.…

In this paper, we investigate the reverse improvement property of Sobolev inequalities on manifolds with quadratically decaying Ricci curvature. Specifically, we establish conditions under which the uniform decay of the heat kernel implies…

In this paper, we prove the compactness theorem for gradient Ricci solitons. Let $(M_{\alpha}, g_{\alpha})$ be a sequence of compact gradient Ricci solitons of dimension $n\geq 4$, whose curvatures have uniformly bounded $L^{\frac{n}{2}}$…

In this note, we prove global weighted Sobolev inequalities on non-compact CD(0,N) spaces satisfying a suitable growth condition, extending to possibly non-smooth and non-Riemannian structures a previous result by V. Minerbe stated for…

We study the qualitative stability of two classes of Sobolev inequalities on Riemannian manifolds. In the case of positive Ricci curvature, we prove that an almost extremal function for the sharp Sobolev inequality is close to an extremal…

This work is devoted to the geometric analysis of metric-measure spaces satisfying a Prekopa-Leindler or a more general Borell-Brascamp-Lieb inequality. Completing the early investigations by Cordero-Erausquin, McCann and Schmuckenschlager,…

In this article, we investigate the geometry of $4$-dimensional compact gradient Ricci solitons. We prove that, under an upper bound condition on the range of the potential function, a $4$-dimensional compact gradient Ricci soliton must…