arXiv research

Minimal volume entropy vanishes for mapping tori over 3-manifolds.

The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.

Extended characterization of RAAGs with zero minimal volume entropy.

Entropy rigidity proven for 3D and higher convex projective manifolds.

Entropy derived from Colding's volume on Ricci-flat manifolds.

Minimal volume entropy vanishes or is positive under certain fiber growth conditions.

Study bounds self-shrinker entropy using Li-Yau volume and Colding-Minicozzi entropy.

Counterexamples found for volume entropy conjecture in hyperbolic 3-manifolds.

We introduce the volume entropy semi-norm in real homology and show that it satisfies functorial properties similar to the ones of the simplicial volume. Answering a question of M. Gromov, we prove that the volume entropy semi-norm is equivalent to the simplicial volume semi-norm in every dimension. We also establish a…

The paper proves a quantitative rigidity result for spaces with specific curvature bounds.

Improved bounds linking entropy and volume in hyperbolic 3-manifolds.

We extend the theory of Patterson-Sullivan measure to any regular covering of a compact manifold using the Busemann compactification and derive an integral formula for the volume entropy. As applications we prove some rigidity theorems for the volume entropy.

Study simplicial volume for fixed fundamental groups, finding gaps.

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 curvature which substantially improves an earlier estimate in \cite{Au2} and giv…

We prove the existence of manifolds with almost maximal volume entropy which are not hyperbolic.

Study minimal volume entropy for free-by-cyclic groups and 2D right-angled Artin groups.

Characterizes metrics with finite total Q-curvature and introduces new volume entropy.

Paper compares two entropy concepts for finite presentation groups.

Investigates properties of volume, entropy, and diameter in higher Teichmüller spaces.

Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.

Optimal geometric estimates for Kähler manifolds with bounded Nash entropy

We consider the volume entropy of closed flat surfaces of genus $g\geq 2$ and area 1. We show that a sequence of flat surfaces diverges in the moduli space if and only if the volume entropy converges to infinity. Equivalently the Hausdorff dimension of the Gromov boundary of the isometric universal cover tends to infin…

This paper proves a curvature entropy inequality for non-symmetric convex bodies.

We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positi…

In this note we give a short proof to the rigidity of volume entropy. The result says that for a closed manifold with Ricci curvature bounded from below, if the universal cover has maximal volume entropy, then it is the space form. This theorem was first proved by F. Ledrappier and X. Wang in [1].

This article deals with topological assumptions under which the minimal volume entropy of a closed manifold, and more generally of a finite simplicial complex, vanishes or is positive. In the first part of the article, we present complementing topological conditions expressed in terms of the growth of the fundamental g…

For a closed, strictly convex projective manifold of dimension $n\geq 3$ that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to…

Estimates open sets for fibrations, leading to volume vanishing results.

The paper studies Q-curvature and volume entropy on manifolds, proving polynomial growth polyharmonic function finiteness and rigidity.

Study minimal surfaces in complex hyperbolic space, linking entropy and volume.

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 their fundamental groups.

The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{po…

We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. We also give …

Harmonic manifolds of hypergeometric type have entropy bounds related to real hyperbolic spaces.

We calculate the volume entropy of local Hermitian symmetric spaces of noncompact type in terms of its invariant $r$, $a$, $b$.

The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.

The study shows almost maximal volume entropy rigidity for certain manifolds with integral Ricci curvature.

Quantitative rigidity theorem for Alexandrov spaces with curvature bounds.

We derive the entropy formula for the linear heat equaiton on complete Riemannian manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman's recent results o…

Sharp bounds for spanning tree entropy in planar lattices.

In this paper it is proven that the volume entropy of a riemannian metric evolving by the Ricci flow, if does not collapse, nondecreases. Therefore, it provides a sufficient condition for a solution to collapse. Then, for the limit solutions of type I or III, the limit entropy is the limit of the entropy as $t$ approac…

We prove that, among metrics on a compact quotient of $\left(\mathbb{H}^2\right)^n$ (product of hyperbolic planes) of prescribed total volume, the product of hyperbolic metrics has minimal volume entropy.

Formula derived for volume entropy of certain metrics on Euclidean space.

Compactness theorem for manifolds with scalar curvature and entropy bounds.

We consider the minimal entropy problem, namely the question of whether there exists a smooth metric of minimal entropy, for certain classes of 3-manifolds. Among other resulsts, we show that if M is a closed, orientable, geometrizable 3-manifold with zero simplicial volume, then the minimal entropy can be solved for M…

We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^2\leq K_X \leq -1$, then $Ent_{top}(X) = n-1$ if and only if $X$ is hyperbolic. In particular, if $X$ has the same length spectrum of a hyperbolic manifold $X_0$, the it is isometric to $X_0$ (we a…

Paper proves volume growth estimate for steady gradient Ricci solitons.

Hyperbolic manifolds are stable under volume-preserving metrics.