Related papers: A sharp Sobolev inequality on Riemannian manifolds

In this note we give a new proof of a version of the Besicovitch covering theorem, given in \cite{EG1992}, \cite{Bogachev2007} and extended in \cite{Federer1969}, for locally finite Borel measures on finite dimensional complete Riemannian…

In this paper we study 4-dimensional $(m,\rho)$-quasi-Einstein manifolds with harmonic Weyl curvature when $m\notin\{0,\pm1,-2,\pm\infty\}$ and $\rho\notin\{\frac{1}{4},\frac{1}{6}\}$. We prove that a non-trivial $(m,\rho)$-quasi-Einstein…

In this paper we consider complete noncompact Riemannian manifolds $(M, g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces $\partial…

We show that if a compact connected $n$-dimensional manifold $M$ has a conformal class containing two non-homothetic metrics $g$ and $\tilde g=e^{2\varphi}g$ with non-generic holonomy, then after passing to a finite covering, either $n=4$…

We establish that any affine manifold $(M,\nabla)$ endowed with a parallel volume form $\omega,$ admits, in any conformal class of Riemannian metrics, a representative $H$ for which $\nabla$ is the Levi-Civita connection. This provides a…

The Schouten tensor \ $A$ \ of a Riemannian manifold \ $(M,g)$ provides important scalar curvature invariants $\sigma_k$, that are the symmetric functions on the eigenvalues of $A$, where, in particular, $\sigma_1$ \ coincides with the…

In this paper we consider Riemannian manifolds of dimension at least $3$, with nonnegative Ricci curvature and Euclidean Volume Growth. For every open bounded subset with smooth boundary we establish the validity of an optimal Minkowski…

In this note, we prove the Bogomolov's inequality over a reduced, compact, irreducible, K\"ahler complex space that is smooth in codimension 2. The proof is obtained by a reduction to the smooth case, using Hironaka's resolution of…

This paper is devoted to establish a class of sharp Sobolev inequalities on the unit complex sphere as follows: 1) Case $0<d<Q=2n+2$: for any $f\in C^\infty$ and $2\leq q \leq \frac{2Q}{Q-d}$, \begin{equation*} \|f\|_q^2\leq…

We present several rigidity results for Riemannian manifolds $(M^n,g)$ with scalar curvature $S \ge -n(n-1)$ (or $S\ge 0$), and having compact boundary $N$ satisfying a related mean curvature inequality. The proofs make use of results on…

We prove that on any compact manifold $M^n$ with boundary, there exist a conformal class $C$ such that for any riemannian metric $g\in C$, $\lambda_1(M^n,g)Vol(M^n,g)^{2/n}< n.Vol(S^n,g_{\textrm{can}})^{2/n}$ and $\sigma_1(M,g,\rho)\mathcal…

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…

Let M be a compact manifold equipped with a Riemannian metric g and a spin structure \si. We let $\lambda (M,[g],\si)= \inf_{\tilde{g} \in [g]} \lambda_1^+(\tilde{g}) Vol(M,\tilde{g})^{1/n}$ where $\lambda_1^+(\tilde{g})$ is the smallest…

A result of M. Ledoux is that a complete Riemannian manifold with non negative Ricci curvature satisfying the Euclidean Sobolev inequality is the Euclidean space. We present a shortcut of the proof. We also give a refinement of a result of…

We review recent results regarding the problem of the stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds. We shall describe techniques and methods from smooth and non-smooth geometry, the fruitful…

We prove a formula for the ${\mathbb S}_n$-equivariant Euler characteristic of the moduli space of graphs $\mathcal{MG}_{g,n}$. Moreover, we prove that the rational ${\mathbb S}_n$-invariant cohomology of $\mathcal{MG}_{g,n}$ stabilizes for…

We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\nabla^{(2)} u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subseteq {\bf R}^n$ and $n\ge 2$, $u:\Omega\rightarrow…

In a 2013 paper, Gromov proves that if smooth Riemannian metrics $g_i$ converge to a smooth Riemannian metric $g$ uniformly, and $g_i$ have scalar curvature uniformly bounded below, then $g$ shares the same scalar curvature lower bound. In…

We consider the second best constant in the Hardy-Sobolev inequality on a Riemannian manifold. More precisely, we are interested with the existence of extremal functions for this inequality. This problem was tackled by Djadli-Druet [5] for…

We prove a sharp Log-Sobolev inequality for submanifolds of a complete non-compact Riemannian manifold with asymptotic non-negative intermediate Ricci curvature and Euclidean volume growth. Our work extends a result of Dong-Lin-Lu which…