Related papers: On Isosystolic Inequalities for T^n, RP^n, and M^3

We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the $1$-skeleton gives an…

Theorem A. Let $M^n$ denote a closed Riemannian manifold with nonpositive sectional curvature and let $\tilde M^n$ be the universal cover of $M^n$ with the lifted metric. Suppose that the universal cover $\tilde M^n$ contains no totally…

Gromov's universal filling inequalities relate the filling radius and the filling volume of a Riemannian manifold to its volume. The main result of the present article is that in dimensions at least three the optimal constants in the…

Let $M^n$ be a closed Riemannian manifold. Larry Guth proved that there exists $c(n)$ with the following property: if for some $r>0$ the volume of each metric ball of radius $r$ is less than $({r\over c(n)})^n$, then there exists a…

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…

Let $(M,g)$ be an $n-$dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying $\rho\Delta_g \rho \geq n- 5\geq 0$, where $\Delta_g$ is the Laplace-Beltrami operator on $(M,g)$ and $\rho$ is the distance…

Given a closed manifold M, we prove the upper bound of (n+d)/2 for the length of a product of systoles that can form a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov's systolic inequalities. Here n is the…

We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of…

Let N be a compact, orientable hyperbolic 3-manifold with connected, totally geodesic boundary of genus 2. If N has Heegaard genus at least 5, then its volume is greater than 6.89. The proof of this result uses the following dichotomy:…

No power law systolic freedom is possible for the product of mod $2$ systoles of dimension $1$ and codimension $1$. This means that any closed $n$-dimensional Riemannian manifold $M$ of bounded local geometry obeys the following systolic…

We show that any closed hyperbolic 3-manifold M admits a Riemannian metric with scalar curvature at least -6, but with volume entropy strictly larger than 2. In particular, this construction gives counterexamples to a conjecture of I. Agol,…

We establish isosystolic inequalities for a class of manifolds which includes the aspherical manifolds. In particular, we relate the systolic volume of aspherical manifolds first to their minimal entropy, then to the algebraic entropy of…

Let $M$ be a Riemannian manifold with dimension greater or equal to $3$ which admits a complete, finite-volume Riemannian metric $g_0$ locally isometric to a rank-1 symmetric space of non-compact type. The volume entropy rigidity theorem…

We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M,g) that are C^0-asymptotic to Schwarzschild of mass m>0. Refining an argument due to H. Bray we obtain an effective volume comparison theorem in…

In this note we prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) An almost flat manifold $(M,g)$ must be flat if it is Einstein, i.e. $\operatorname{Ric}_g=\lambda g$ for some real number $\lambda$.…

We study the size of the isometry group Isom(M, g) of Riemannian manifolds (M, g) as g varies. For M not admitting a circle action, we show that the order of Isom(M, g) can be universally bounded in terms of the bounds on Ricci curvature,…

The goal of this article is to study the space of smooth Riemannian structures on compact manifolds with boundary that satisfies a critical point equation associated with a boundary value problem. We provide an integral formula which…

Let $(M,g)$ be a $C^\infty$-smooth, $n$-dimensional Riemannian manifold which is diffeomorphic to $\RR^n$ and admit an action of a properly discontinuous and cocompact group. This work proves the existence of a $C^\infty$ equivariant…

According to Mostow's celebrated rigidity theorem, the geometry of closed hyperbolic 3-manifolds is already determined by their topology. In particular, the volume of such manifolds is a topological invariant and, as such, has been…

If M is a closed simple 3-manifold whose fundamental group contains a genus-g surface group for some g>1, and if the dimension of H_1(M;Z_2) is at least max(3g-1,6), we show that M contains a closed, incompressible surface of genus at most…