Related papers: Maximal volume entropy rigidity for $\mathsf{RCD}^…

Consider a sequence of Riemannian manifolds $(M^n_i,g_i)$ with scalar curvatures and entropies bounded below by small constants $R_i,\mu_i \geq-\epsilon_i$. The goal of this paper is to understand notions of convergence and the structure of…

In this paper we prove that under a lower bound on the Ricci curvature and an asymptotic assumption on the scalar curvature, a complete conformally compact manifold $(M^{n+1},g)$, with a pole $p$ and with the conformal infinity in the…

We prove two rigidity results for complete Riemannian three-manifolds of higher rank. Complete three-manifolds have higher spherical rank if an only if they are spherical space forms. Complete finite volume three-manifolds have higher…

We give new estimates on the lower bounds for the first closed or Neumann eigenvalue for a compact manifold with positive Ricci curvature in terms of the diameter and the lower bound of Ricci curvature. The results improve the previous…

The aim of this note is to generalize to the class of non collapsed RCD(K,N) metric measure spaces the volume bound for the effective singular strata obtained by Cheeger and Naber for non collapsed Ricci limits in \cite{CheegerNaber13a}.…

Let $X$ be a compact Gromov-Hausdorff limit space of a collapsing sequence of compact $n$-manifolds, $M_i$, of Ricci curvature $\text{Ric}_{M_i}\ge -(n-1)$ and all points in $M_i$ are $(\delta,\rho)$-local rewinding Reifenberg points, or…

In this paper, we develop the infinitesimal geometry of the limit spaces of compact Riemannian manifolds with boundary, where we assume lower bounds on the sectional curvatures of manifolds and boundaries and the second fundamental forms of…

Given a complete isometric immersion $\phi: P^m \longrightarrow N^n$ in an ambient Riemannian manifold $N^n$ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially…

In [16], we established Trudinger-Moser inequalities for complete noncompact Riemannian manifold on which the Ricci curvature has lower bound and the injectivity radius is strictly positive. In this note, we improve those inequalties when…

In this paper, we present finite topological type theorems for open manifolds with non-negative Ricci curvature, under almost maximal local rewinding volume. Unlike previous related research, our theorems remove the constraints of sectional…

In this paper, we first prove a compactness theorem for the space of closed embedded $f$-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-\'{E}mery Ricci curvature. Then we give a Lichnerowicz type lower…

Let $(M,g)$ be a $3$--dimensional, complete, one--ended Riemannian manifold, with a minimal, compact and connected boundary. We assume that $M$ has a simple topology and that the scalar curvature of $(M,g)$ is non--negative. Moreover, we…

Using the localization technique, we prove a sharp upper bound on the first Dirichlet eigenvalue of metric balls in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces. This extends a celebrated result of Cheng to the non-smooth…

In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by $4\pi/c$,…

By using optimal mass transport theory we prove a sharp isoperimetric inequality in ${\sf CD} (0,N)$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for…

We study the rigidity of compact submanifolds of Riemannian manifolds of arbitrary codimension that satisfy a sharp pinching condition involving the norm of the second fundamental form and the mean curvature. Without assuming that the…

We prove a finiteness theorem for the class of complete finite volume Riemannian manifolds with pinched negative sectional curvature, fixed fundamental group, and of dimension $>2$. One of the key ingredients is that the fundamental group…

We shall prove that under some volume growth condition, the essential spectrum of the Laplacian contains the interval $[(n-1)^2K/4, \infty)$ if an $n$-dimensional Riemannian manifold has an end and the average of the part of the Ricci…

We prove an upper bound on the rank of the abelianised revised fundamental group (called "revised first Betti number") of a compact $RCD^{*}(K,N)$ space, in the same spirit of the celebrated Gromov-Gallot upper bound on the first Betti…

We study Serrin's overdetermined boundary value problems in bounded domains on weighted Riemannian manifolds. When the closure of the domain is compact, we establish a rigidity result that characterizes both the solution and the geometry of…