arXiv research

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

Upper bound for Laplacian eigenvalue via conformal volume.

Upper bound on geodesic ball volume in Riemannian manifolds.

New examples show no upper bounds on link volumes on incompressible surfaces.

Upper bounds for volumes of hyperbolic polyhedra and links are derived.

Improved algorithm for modular links provides upper volume bounds.

In this note we provide several lower bounds for the volume of a geodesic ball within the injectivity radius in a $3$-dimensional Riemannian manifold assuming only upper bounds for the Ricci curvature.

We give a characterization of critical points that allows us to define a metric invariant on all Riemannian manifolds $M$ with a lower sectional curvature bound and an upper radius bound. We show there is a uniform upper volume bound for all such manifolds with an upper bound on this invariant. We generalize results by…

We use spectral embeddings to give upper bounds on the spectral function of the Laplace--Beltrami operator on homogeneous spaces in terms of the volume growth of balls. In the case of compact manifolds, our bounds extend the 1980 lower bound of Peter Li for the smallest positive eigenvalue to all eigenvalues. We also i…

A periodic geodesic on a surface has a natural lift to the unit tangent bundle; when the complement of this lift is hyperbolic, its volume typically grows as the geodesic gets longer. We give an upper bound for this volume which is linear in the geometric length of the geodesic.

We give a refined upper bound for the hyperbolic volume of an alternating link in terms of the first three and the last three coefficients of its colored Jones polynomial.

We prove a so called $κ$ 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 opposite statement of Perelman's $κ$ non-collapsing property for Ricci flow. These two resul…

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold such that $π_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that strict upper bounds of 0.07625, 0.1525 and 0.22875 for ${\rm vol}\ {\mathfrak M}$ imply respective upper bounds of 23, 43 and 79 for $\dim H_1({\mathfrak M};{\mathbb F}_2…

Upper bounds for Steklov eigenvalues of warped products are derived.

Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. We study the geometry of twisted torus links and related generalizations. We determine upper bounds on their …

In this paper we generalize the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimizing closed hypersurface $Σ$ of a Riemannian 5-manifold $M$ with scalar curvature bounded from below by a positive constant in terms of the total…

Upper bound on 3-manifold volumes from surface homeomorphisms.

Upper bounds for Steklov eigenvalues derived from intersection indices.

Using $δ$-invariants and Newton--Okounkov bodies, we derive the optimal volume upper bound for Kähler manifolds with positive Ricci curvature, from which we get a new characterization of the complex projective space.

Upper bounds for essential spectrum of minimal submanifolds linked to volume growth.

Study bounds Neumann and Steklov eigenvalues on manifolds and submanifolds.

Estimates volume of convex Alexandrov spaces with boundary.

This paper sets out to provide a general framework for the pricing of average-type options via lower and upper bounds. This class of options includes Asian, basket and options on the volume-weighted average price. We demonstrate that in cases under discussion lower bounds allow for the dimensionality of the problem to …

The smallest $r$ so that a metric $r$-ball covers a metric space $M$ is called the radius of $M$. The volume of a metric $r$-ball in the space form of constant curvature $k$ is an upper bound for the volume of any Riemannian manifold with sectional curvature $\geq k$ and radius $\leq r$. We show that when such a manifo…

We utilize ideal bipyramids to obtain new upper bounds on volume for hyperbolic link complements in terms of the combinatorics of their projections.

Upper and lower bounds for hyperbolic rod complements in 3-torus volumes.

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…

Bounding characteristic numbers of Riemannian manifolds via volume.

Compactness theorem for manifolds with scalar curvature and entropy bounds.

We obtain upper bounds for the eigenvalues of the Schrödinger operator $L=Δ_g+q$ depending on integral quantities of the potential $q$ and a conformal invariant called the min-conformal volume. Moreover, when the Schrödinger operator $L$ is positive, integral quantities of $q$ which appear in upper bounds, can be repla…

We find upper and lower bounds for the first eigenvalue and the volume entropy of a noncompact real analytic Kähler manifold, in terms of Calabi's diastasis function and diastatic entropy, which are sharp in the case of the complex hyperbolic space. As a corollary we obtain explicit lower bounds for the first eigenvalu…

Torus covers have controlled volume and diameter under curvature and diameter bounds.

Paper simplifies link classification in 3-sphere using braids and templates.

Upper bounds for Steklov eigenvalues on manifolds with boundary.

Bray's football theorem (\cite{bray2009penrose}) is a weakening of Bishop theorem in dimension 3. It gives a sharp volume upper bound for a three dimensional manifold with scalar curvature larger than $n(n-1)$ and Ricci curvature larger than $\varepsilon$. This paper extends Bray's football theorem in high dimensions, …

We show that integral foliated simplicial volume of closed manifolds gives an upper bound for the cost of the corresponding fundamental groups.

For a compact right-angled polyhedron $R$ in $\mathbb H^3$ denote by $\operatorname{vol} (R)$ the volume and by $\operatorname{vert} (R)$ the number of vertices. Upper and lower bounds for $\operatorname{vol} (R)$ in terms of $\operatorname{vert} (R)$ were obtained in \cite{A09}. Constructing a 2-parameter family of po…

We observe that stable integral simplicial volume of closed manifolds gives an upper bound for the rank gradient of the corresponding fundamental groups.

The study of systoles in arithmetic hyperbolic manifolds.

The paper sets new limits on hyperbolic polyhedra volumes.

For families of knots and links given in Conway notation we compute lower maximal and upper minimal bound of hyperbolic volume by using source links and augmented links.

Optimizes bounds for threefold singularity volumes.

Upper bounds on nullhomotopy volumes in nilpotent spaces are refined.

We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good 3-orbifolds with planar singular locus and underlying manifold the 3-sphere. The volume b…

We give an upper bound for the growth of homology torsions of finite coverings of irreducible 3-manifolds with tori boundary in terms of hyperbolic volume.

New bounds on Laplace operator eigenvalues for submanifolds in Euclidean spaces.

The paper proves macroscopic versions of conjectures about scalar curvature and volume bounds.

We examine volume pinching problems of CAT(1) spaces. We characterize a class of compact geodesically complete CAT(1) spaces of small specific volume. We prove a sphere theorem for compact CAT(1) homology manifolds of small volume. We also formulate a criterion of manifold recognition for homology manifolds on volume g…