Related papers: Volume of the domain bounded by a Hermitian quadri…

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…

Let M either be a closed real analytic Riemannian manifold or a closed smooth Riemannian surface. We estimate from below the volume of a nodal domain component in an arbitrary ball provided that this component enters the ball deeply enough.

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.

Associated to a holomorphic quadratic differential is a unit ball of the measured lamination space. The Thurston volume of the unit ball defines a function on the moduli space. We show that the volume function is not proper and characterize…

We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different…

We derive an explicit formula for the volume of a regular simplex in the hyperbolic space of any dimension.

We consider the projectivization of Minkowski space with the analytic continuation of the hyperbolic metric and call this an extended hyperbolic space. We can measure the volume of a domain lying across the boundary of the hyperbolic space…

In the context of Synthetic Differential Geometry, we describe the square volume of a ``second-infinitesimal simplex'', in terms of square-distance between its vertices. The square-volume function thus described is symmetric in the…

Heron's formula from antiquity, for the area of a triangle, is used to relate volume form and infinitesimal square-volume of certain infinitesimal simplices in a Riemannian manifold

We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient…

Let $\mathcal{A}_g$ denote the moduli stack of principally polarized abelian varieties of dimension $g$. The arithmetic height, or arithmetic volume, of $\overline{\mathcal{A}}_g$, is defined to be the arithmetic degree of the metrized…

We quantise complex, infinite-dimensional projective space CP(H). We apply the result to quantise a complex, finite-dimensional, classical phase space C whose symplectic volume is infinite, by holomorphically embedding it into CP(H). The…

Let $X$ be a smooth projective toric variety over $Spec(Z)$. Let $\bar{\mathcal{O}(D)}$ be an equivariant hermitian line bundle on $X$, equipped with a positive metric and invariant under the action of the compact torus of $X(C)$, we…

We establish formulas that give the intrinsic volumes, or curvature measures, of sublevel sets of functions defined on Riemannian manifolds as integrals of functionals of the function and its derivatives. For instance, in the Euclidean…

The period is a classical complex analytic invariant for a compact Riemann surface defined by integration of differential 1-forms. It has a strong relationship with the complex structure of the surface. In this chapter, we review another…

The purpose of this article is to investigate the geometry of the set of locally diagonalizable bipartite quantum states. We have the following new results: the Hilbert-Schmidt volume of all locally diagonalizable states, and a necessary…

Estimating the volume of a convex body is a canonical problem in theoretical computer science. Its study has led to major advances in randomized algorithms, Markov chain theory, and computational geometry. In particular, determining the…

Volume of metric balls relates to rate-distortion theory and packing bounds on codes. In this paper, the volume of balls in complex Grassmann manifolds is evaluated for an arbitrary radius. The ball is defined as a set of hyperplanes of a…

There exists, in general, no unique definition of the size (volume, area, etc., depending on dimension) of a soliton. Here we demonstrate that the geometric volume (area etc.) of a soliton is singled out in the sense that it exactly…

We prove that the largest first eigenvalue of the Dirac operator among all hermitian metrics on the complex projective space of odd dimension $m$, larger than the Fubini-Study metric is bounded by $(2m(m+1))^{1/2}$.