Related papers: Differentiable Rigidity under Ricci curvature lowe…

In this work, we study several inequalities related to a Dirichlet problem on Riemannian manifolds whose Ricci curvature is bounded from below. First, we establish inequalities involving the torsional rigidity and discuss rigidity results…

We review recent results on the study of the isoperimetric problem on Riemannian manifolds with Ricci lower bounds. We focus on the validity of sharp second order differential inequalities satisfied by the isoperimetric profile of possibly…

This is a continuation of the research in [16]. Let $(\overline{M},g_{-1})$ be a closed geodesic $r_0$-ball in the hyperbolic space $(\mathbb{H}^n,g_{-1})$. Let $m\neq1$ be a positive constant. In this paper, we show that for $n\geq3$,…

Let $(M^n,g)$ be a complete Riemannian manifold which is not isometric to $\mathbb{R}^n$, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set $\mathcal{G}\subset…

In this paper we prove weak L^{1,p} (and thus C^{\alpha}) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We…

This is the second paper of two in a series under the same title ([CRX]); both study the quantitative volume space form rigidity conjecture: a closed $n$-manifold of Ricci curvature at least $(n-1)H$, $H=\pm 1$ or $0$ is diffeomorphic to a…

In this paper, we investigate local rigidity properties related to Gagliardo-Nirenberg constants and unweighted Yamabe-type constants. Let $V$ be an open bounded subset of an $n$-dimensional Riemannian manifold $(M,g)$ whose…

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 existence of isoperimetric regions for every volume in non-compact Riemannian $n$-manifolds $(M,g)$, $n\geq 2$, having Ricci curvature $Ric_g\geq (n-1) k_0 g$ and being locally asymptotic to the simply connected space form of…

In Riemannian geometry, the Cheng's maximal diameter rigidity theorem says that if a complete $n$-manifold $M$ of Ricci curvature, $\operatorname{Ric}_M\ge (n-1)$, has the maximal diameter $\pi$, then $M$ is isometric to the unit sphere…

We show that a complete Riemannian manifold of dimension $n$ with $\Ric\geq n{-}1$ and its $n$-st eigenvalue close to $n$ is both Gromov-Hausdorff close and diffeomorphic to the standard sphere. This extends, in an optimal way, a result of…

We show two stability results for a closed Riemannian manifold whose Ricci curvature is small in the Kato sense and whose first Betti number is equal to the dimension. The first one is a geometric stability result stating that such a…

Let $(M, g)$ be a complete, connected, non-compact Riemannian $3$-manifold. Suppose that $(M,g)$ satisfies the Ricci--pinching condition $\mathrm{Ric}\geq\varepsilon\mathrm{R} g$ for some $\varepsilon>0$, where $\mathrm{Ric}$ and…

Recall that the radius of a compact metric space $(X, dist)$ is given by $rad\ X = \min_{x\in X} \max_{y\in X} dist(x,y)$. In this paper we generalize Berger's $\frac{1}{4}$-pinched rigidity theorem and show that a closed, simply connected,…

In this paper, we are going to show some rigidity results for complete open Riemannian manifolds with nonnegative scalar curvature. Without using the famous Cheeger-Gromoll splitting theorem we give a new proof to a rigidity result for…

We consider complete (possibly non-compact) three dimensional Riemannian manifolds (M,g) such that: a) (M,g) is non-collapsed, b) the Ricci curvature of (M,g) is bounded from below, c) the geometry of (M,g) at infinity is not too extreme.…

A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin. The argument makes use of a quasi-local mass characterization of Euclidean balls…

The inscribed radius of a compact manifold with boundary is bounded above if its Ricci curvature and mean curvature are bounded from below. The rigidity result implies that the upper bound can be achieved only in space form. In this paper,…

The sharp isoperimetric inequality for non-compact Riemannian manifolds with non-negative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions…

The lens data of a Riemannian manifold with boundary is the collection of lengths of geodesics with endpoints on the boundary together with their incoming and outgoing vectors. We show that negatively-curved Riemannian manifolds with…