arXiv research

For an equiregular sub-Riemannian manifold M, Popp's volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp's volume, written in terms of a frame adapted to the sub-Riemannia…

Upper bound on geodesic ball volume in Riemannian manifolds.

Positive simplicial volume implies locally symmetric space structure.

Upper bounds for volume spectrum depend on volume, dimension, and a conformal invariant.

Ancient formula connects volume forms and infinitesimal square volumes in manifolds.

Paper proves a new volume comparison theorem for Riemannian manifolds.

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

Study on volume of tubes and concentration in Riemannian geometry.

Prove a generalization of Werner's formula for the volume of illumination bodies on Riemannian manifolds.

Proves spectra equivalence for Riemannian manifolds.

Sharp heat kernel and Green's function estimates on Euclidean volume growth manifolds.

Study approximates sub-Riemannian structures with Riemannian metrics and analyzes spectral convergence.

In this paper, we will count the number of cusps of complete Riemannian manifolds $M$ with finite volume. When $M$ is a complete smooth metric measure spaces, we show that the number of cusps in bounded by the volume $V$ of $M$ if some geometric conditions hold true. Moreover, we use the nonlinear theory of the $p$-Lap…

Sharp volume growth ratio for 3D manifolds with positive scalar curvature.

Compactness theorem for Riemannian manifolds with volume and curvature bounds.

Study Liouville equation on Riemannian surfaces, linking volume growth to classification results.

Sub-Riemannian spectral distance defined using eigenfunctions of sub-Laplacian

Upper bound for Laplacian eigenvalue via conformal volume.

New proof shows affine manifolds with parallel volume are Riemannian-flat.

The aim of this short note is to produce new examples of geometrical flows associated to a given Riemannian flow $g(t)$. The considered flow in covariant symmetric $2$-tensor fields will be called Ricci-Yamabe map since it involves a scalar combination of Ricci tensor and scalar curvature of $g(t)$. Due to the signs of…

Weyl's intrinsic volumes converge to the Euler characteristic of the base manifold under certain metrics.

We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a c…

Integral of scalar curvature equals a volume ratio term on certain 3D manifolds.

We relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular at…

We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure

If (M^n, g) is a complete Riemannian manifold with filling radius at least R, then we prove that it contains a ball of radius R and volume at least c(n)R^n. If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most c(n)Volume(M,hyp), then we prove that the universal cover of (M,g…

Bounding characteristic numbers of Riemannian manifolds via volume.

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, which is extensively discussed. For the general semi-Riemannian case, a local v…

Proves rigidity for maps between manifolds using degree theory and current developments.

Rough and Hodge Laplacians eigenvalues approach zero with fixed volume.

The simplicial volume of non-R^3 contractible 3-manifolds is infinite.

New Crofton formulae derived from existing ones.

Estimates volume of convex Alexandrov spaces with boundary.

Derives a new maximum principle for Riemannian manifolds with volume growth constraints.

We provide some criteria to $p$-parabolicity of Riemannian submersions. In particular, if $N$ is $p$-parabolic and $π:M\to N$ is a Riemannian submersion with uniformly bounded volume of fibers, then $M$ is also $p$-parabolic. In the case of warped manifolds we characterize $p$-parabolicity in terms of a volume growth c…

The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.

For any closed smooth Riemannian manifold H. Weyl has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of th…

In this paper we explain how to define "lower dimensional'' volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential …

The study shows conditions for larger volumes in the universal cover of a manifold.

If $(M^n, g)$ is a closed Riemannian manifold where every unit ball has volume at most $ε_n$ (a sufficiently small constant), then the $(n-1)$-dimensional Uryson width of $(M^n, g)$ is at most 1.

The paper establishes a sub-additive inequality for volume and ε-phase-transition spectra of Riemannian manifolds.

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, whil…

We study the variation of a smooth volume form along extremals of a variational problem with nonholonomic constraints and an action-like Lagrangian. We introduce a new invariant describing the interaction of the volume with the dynamics and we study its basic properties. We then show how this invariant, together with c…

New metric properties show volume constraints in collapsing spaces.

The paper studies sub-Riemannian geometry and proves a Weyl's invariance result for Heisenberg groups.

Derives Weyl law for volume spectrum using parametric inequalities.

Study shows shortest geodesic length on certain manifolds is limited by volume, diameter, and cover elements.

Volume is a natural measure of complexity of a Riemannian manifold. In this survey, we discuss the results and conjectures concerning n-dimensional hyperbolic manifolds and orbifolds of small volume.