Related papers: Enhanced Bishop-Gromov Theorem

We prove that any gradient shrinking Ricci soliton has at most Euclidean volume growth. This improves a recent result of H.-D. Cao and D. Zhou by removing a condition on the growth of scalar curvature.

Suppose that a countable group $G$ admits a cusp-uniform action on a hyperbolic space $(X,d)$ such that $G$ is of divergent type. The main result of the paper is characterizing the purely exponential growth type of the orbit growth function…

The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank~1. This conjecture has been proved by Z. Szab\'{o} \cite{Sz} for harmonic manifolds with compact universal cover. E. Damek…

In this article we study geodesic flows on closed Riemannian manifolds without conjugate points and divergence property of geodesic rays. If the fundamental group is Gromov hyperbolic and residually finite we prove, under appropriate…

We study the geodesic flow of geometrically finite quotients $\Omega/{\Gamma}$ of Hilbert geometries, in particular its recurrence properties. We prove that, under a geometrical assumption on the cusps, the geodesic flow is uniformly…

We establish an abstract local ergodic theorem, under suitable space-time scaling, for the (boundary-driven) symmetric exclusion process on an increasing sequence of balls covering an infinite weighted graph. The proofs are based on 1-block…

Let $f$ be a $C^{1}$ diffeomorphism on a compact manifold $M$ admitting a partially hyperbolic splitting $TM=E^{s}\oplus_{\prec} E^{1}\oplus_{\prec} E^{2}\cdots \oplus_{\prec}E^{l}\oplus_{\prec} E^{u}$ where $E^{s}$ is uniformly…

We investigate the interaction between systolic geometry and positive scalar curvature through spinorial methods. Our main theorem establishes an upper bound for the two-dimensional stable systole on certain high-dimensional manifolds with…

In this paper, we prove that for any given closed contact manifold, there exists an infinite-dimensional space of Riemannian metrics which can be identified with the space of bundle metrics on the induced contact distribution. For each such…

Given a Riemannian metric on a homotopy $n$-sphere, sweep it out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as…

We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show…

In this article, we consider the geodesic flow on a compact rank $1$ Riemannian manifold $M$ without focal points, whose universal cover is denoted by $X$. On the ideal boundary $X(\infty)$ of $X$, we show the existence and uniqueness of…

Lorentzian versions of classical Riemannian volume comparison theorems by Gunther, Bishop and Bishop-Gromov, are stated for suitable natural subsets of general semi-Riemannian manifolds. The problem is more subtle in the Bishop-Gromov case,…

A volume growth-based proof of the Multiplicative Ergodic Theorem for Banach spaces is presented, following the approach of Ruelle for cocycles acting on a Hilbert space. As a consequence, we obtain a volume growth interpretation for the…

For an open manifold $M$ and a function $v$ with bounded growth of derivative, there exists a Riemannian metric of bounded geometry on $M$ such that the volume growth function lies in the same growth class as $v$. This was proved by R.…

We consider curvature flows in hyperbolic space with a monotone, symmetric, homogeneous of degree 1 curvature function F. Furthermore we assume F to be either concave and inverse concave or convex. For compact initial hypersurfaces, which…

We demonstrate that there exists a class of cyclic cosmological models, such that these models can in principle solve the problem of the entropy growth, and are at the same time geodesically complete. We thus show that some recently stated…

We obtain an improved pseudolocality result for Ricci flows on two-dimensional surfaces that are initially almost-hyperbolic on large hyperbolic balls. We prove that, at the central point of the hyperbolic ball, the Gauss curvature remains…

In a Riemannian manifold, it is well known that the scalar curvature at a point can be recovered from the volumes (areas) of small geodesic balls (spheres). We show the scalar curvature is likewise determined by the relative capacities of…

We prove the upper semicontinuity of the measure theoretic entropy for the geodesic flow on complete Riemannian manifolds without focal points and bounded sectional curvature. We then study the relationship between the escape of mass…