arXiv research

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

Proof of Tholozan's volume formula for anti-de-Sitter 3-manifolds.

Formula calculates volume of two-bridge knots.

Formula connects foliated simplicial volume with group cost.

New formulas for coassociative submanifolds' volume variation.

New formula calculates volumes of ideal hyperbolic drums.

Exact formulas for volumes of specific knot cone-manifolds.

New Crofton formulae derived from existing ones.

Formula connects curvature to volume in special geometric spaces.

Formula for volumes of odd strata of quadratic differentials using graph intersection numbers.

Harish-Chandra's volume formula shows that the volume of a flag manifold $G/T$, where the measure is induced by an invariant inner product on the Lie algebra of $G$, is determined up to a scalar by the algebraic properties of $G$. This article explains how to deduce Harish-Chandra's formula from Weyl's law by utilizing…

We calculate the volume of the $7_3^2$ link cone-manifolds using the Schläfli formula. As an application, we give the volume of the cyclic coverings over the link.

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…

Given a differentiable deformation of geometrically finite hyperbolic $3$-manifolds $(M_t)_t$, the Bonahon-Schläfli formula expresses the derivative of the volume of the convex cores $(C M_t)_t$ in terms of the variation of the geometry of its boundary, as the classical Schläfli formula does for the volume of hyperboli…

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.

Let $M$ be a compact oriented three-manifold whose interior is hyperbolic of finite volume. We prove a variation formula for the volume on the variety of representations of $M$ in $\operatorname{SL}_n(\mathbb C)$. Our proof follows the strategy of Reznikov's rigidity when $M$ is closed, in particular we use Fuks' appro…

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

The paper analyzes distances and volumes in lens spaces using recursion and formulas.

We derive an analytic formula for the dual Jacobian matrix of a generalised hyperbolic tetrahedron. Two cases are considered: a mildly truncated and a prism truncated tetrahedron. The Jacobian for the latter arises as an analytic continuation of the former, that falls in line with a similar behaviour of the correspondi…

Formula calculates volume of CMC surfaces with translational periods, disproving isoperimetric conjecture.

We provide an isoperimetric inequality for critical metrics of the volume functional with nonnegative scalar curvature on compact manifolds with boundary. In addition, we establish a Weitzenböck type formula for critical metrics of the volume functional on four-dimensional manifolds. As an application, we obtain a clas…

We derive a new renormalized volume formula for conformally compact asymptotically hyperbolic manifolds in dimension four. The formula generalizes the ones given by Anderson, Albin, and Chang-Qing-Yang for the case of Poincare-Einstein manifolds. We also derive variational formulas for the renormalized seen as a functi…

We compute the Riemannian volume on the moduli space of flat connections on a nonorientable 2-manifold, for a natural class of metrics. We also show that Witten's volume formula for these moduli spaces may be derived using Haar measure, and we give a new proof of Witten's volume formula for the moduli space of flat con…

In this note, we answer a question of Mirzakhani on asymptotic behavior of the one-point volume polynomial of moduli spaces of curves. We also present some applications of Mirzakhani's asymptotic formulae of Weil-Petersson volumes.

We discuss Witten's formulas for the symplectic volumes of moduli spaces of flat connections on 2-manifolds from the viewpoint of Hamiltonian cobordism as introduced by Ginzburg-Guillemin-Karshon.

We give an efficient simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic PSL(2,C)-representation of a tame 3-manifold. If the representation is the geometric representation of a hyperbolic 3-manifold, our formula computes the volume and Chern-Simons invariant directly from an ideal tria…

Weyl's tube formula holds for various cross-sections under symmetry conditions.

We present the area and coarea formulas for Lipschitz maps, valid for general volume densities. As applications, we give a short, "euclidean" proof of the anisotropic Sobolev inequality and describe an anisotropic tube formula for hypersurfaces in $mathbb{R}^n$. A discussion about the first variation of the anisotropic…

We prove a Gauss-Bonnet type formula for Riemann-Finsler surfaces of non-constant indicatrix volume and with regular piecewise smooth boundary. We give a Hadamard type theorem for N-parallels of a Landsberg surface.

The paper calculates volumes of moduli spaces of flat metrics on spheres with specific angles.

The paper derives formulas for symplectic volume forms on surface representation varieties.

We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.

Study critical metrics on manifolds with boundary using integral and boundary estimates.

We prove an explicit cabling formula for the colored Jones polynomial. As an application we prove the volume conjecture for all zero volume knots and links, i.e. all knots and links that are obtained from the unknot by repeated cabling and connected sum.

Formula derived for enclosed volume of CMC surfaces in 3-sphere.

Study proves a new formula for capillary hypersurfaces and shows a flow converging to a special shape.

In this short paper, we re-derive the Bochner formula for the Laplacian by considering local variations of volume. The derivation is rooted in the fact that the Laplacian of a function measures the volume variation along the flow of the gradient vector of the function. Possible extensions of this approach/technique are…

A generalized hyperbolic tetrahedra is a polyhedron (possibly non-compact) with finite volume in hyperbolic space, obtained from a tetrahedron by the polar truncation at the vertices lying outside the space. In this paper it is proved that a volume formula for ordinary hyperbolic tetrahedra devised by J. Murakami and M…

The volume of the quantum mechanical state space over $n$-dimensional real, complex and quaternionic Hilbert-spaces with respect to the canonical Euclidean measure is computed, and explicit formulas are presented for the expected value of the determinant in the general setting too. The case when the state space is endo…

We formulate a conjectural Lefschetz formula for locally symmetric spaces of finite volume. The formula can be verified in the compact case and for Riemann surfaces.

We derive a formula of Chern-Gauss-Bonnet type for the Euler characteristic of a four dimensional manifold-with-boundary in terms of the geometry of the Loewner-Nirenberg singular Yamabe metric in a prescribed conformal class. The formula involves the renormalized volume and a boundary integral. It is shown that if the…

A formula of the renormalized volume of tubes over polalized Kähler-Einstein manifolds is given in terms of the Einstein constant and the volume of the polarization.

A new formula approximates knot volume using Jones polynomial evaluations.

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…

Recursion formula derived for moduli spaces of hyperbolic surfaces with cone points.

We consider the volumes of classical supermanifolds such as the supersphere, complex projective superspace, and Stiefel and Grassmann supermanifolds, with respect to the natural metrics or symplectic structures. We show that the formulas for the volumes, upon certain universal normalization, can be obtained by an analy…

Loosely speaking, the Volume Conjecture states that the limit of the n-th colored Jones polynomial of a hyperbolic knot, evaluated at the primitive complex n-th root of unity is a sequence of complex numbers that grows exponentially. Moreover, the exponential growth rate is proportional to the hyperbolic volume of the …

We present two identities (contiguity relation and variation formula) concerning the volume of a spherically faced simplex in the Euclidean space. These identities are described in terms of Cayley-Menger determinants and their differentials involved with hypersphere arrangements. They are derived as a limit of fundamen…