Related papers: Enhanced Bishop-Gromov Theorem

The Bishop-Gromov theorem is a comparison theorem of differential geometry that upperbounds the growth of volume of a geodesic ball in a curved space. For many spaces, this bound is far from tight. We identify a major reason the bound fails…

We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric…

The Bishop-Gromov bound -- a cousin of the focusing lemmas that Hawking and Penrose used to prove their black hole singularity theorems -- is a differential geometry result that upperbounds the rate of growth of volume of geodesic balls in…

In section 1 we reformulate a theorem of Blichfeldt in the framework of manifolds of nonpositive curvature. As a result we obtain a lower bound on the number of homotopically distinct geodesic loops emanating from a common point q whose…

We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For…

We give lower bounds for the growth of the number of Reeb chords and for the volume growth of Reeb flows on spherizations over closed manifolds M that are not of finite type, have virtually polycyclic fundamental group, and satisfy a mild…

We prove a so called $\kappa$ non-inflating property for Ricci flow, which provides an upper bound for volume ratio of geodesic balls over Euclidean ones, under an upper bound for scalar curvature. This result can be regarded as the…

We consider a geodesic flow on a compact manifold endowed with a Riemannian (or Finsler, or Lorentz) metric satisfying some generic, explicit conditions. We couple the geodesic flow with a time-dependent potential, driven by an external…

We study the volume growth of metric balls as a function of the radius in discrete spaces, and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called…

We investigate the effect of the average scalar curvature on the conjugate radius, average area of the geodesic spheres, average volume of the metric balls and the total volume of a closed Riemannian manifold $N$ (or more generally $N$ with…

The regularity of limit spaces of Riemannian manifolds with L^p curvature bounds, $p > n/2$, is investigated under no apriori non-collapsing assumption. A regular subset, defined by a local volume growth condition for a limit measure, is…

We give several Bishop-Gromov relative volume comparisons with integral Ricci curvature which improve the results in \cite{PW1}. Using one of these volume comparisons, we derive an estimate for the volume entropy in terms of integral Ricci…

We prove that almost all geodesics on a noncompact locally symmetric space of finite volume grow with a logarithmic speed -- the higher rank generalization of a theorem of D. Sullivan (1982). More generally, under certain conditions on a…

We provide an overview of technics that lead to an Euclidean upper bound on the volume of geodesic balls.

We examine topological properties of pointed metric measure spaces $(Y, p)$ that can be realized as the pointed Gromov-Hausdorff limit of a sequence of complete, Riemannian manifolds $\{(M^n_i, p_i)\}_{i=1}^{\infty}$ with nonnegative Ricci…

The question of geodesic completeness of cosmological spacetimes has recently received renewed scrutiny. A particularly interesting result is the observation that the well-known Borde-Guth-Vilenkin (BGV) theorem may misdiagnose geodesically…

We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound $l$ for the cut distance and Ricci…

In this paper, we prove the optimal volume growth for complete Riemannian manifolds $(M^n,g)$ with nonnegative Ricci curvature everywhere and bi-Ricci curvature bounded from below by $n-2$ outside a compact set when the dimension is less…

We show a sharp and rigid spectral generalization of the classical Bishop--Gromov volume comparison theorem: if a closed Riemannian manifold $(M,g)$ of dimension $n\geq3$ satisfies $$…

Given a bi-Lipschitz measure-preserving homeomorphism of a compact metric measure space of finite dimension, consider the sequence formed by the Lipschitz norms of its iterations. We obtain lower bounds on the growth rate of this sequence…