arXiv research

Study weakly weighted Einstein-Finsler metrics, showing specific curvature properties and characterizing them.

The paper proposes extensions of the usual notions of Finslerian volume to time orientable Finsler spacetime manifolds. The basic idea is to replace, in the classical Busemann-Hausdorff and Holmes-Thompson definitions, integration on the indicatrices of the given metric (which are, in Lorentzian signature, non-compact,…

Study on Berwald-Weyl curvature with projective invariance and vanishing results.

We explore a connection between the Finslerian area functional based on the Busemann-Hausdorff-volume form, and well-investigated Cartan functionals to solve Plateau's problem in Finsler 3-space, and prove higher regularity of solutions. Free and semi-free geometric boundary value problems, as well as the Douglas probl…

In this paper, the isoperimetric problem in the 2-dimensional Finsler space form $(F_B,B^2(1))$ with k = 0 by using the Busemann-Hausdorff area is investigated. We prove that the circle centered the origin achieves the local maximum area of the isoperimetric problem.

New optimal isosystolic inequality found for Finsler reversible 2-tori.

Given a Finsler manifold $(M,F)$, it is proved that the first eigenvalue of the Finslerian $p$-Laplacian is bounded above by a constant depending on $\ p$, the dimension of $M$, the Busemann-Hausdorff volume and the reversibility constant of $(M,F)$. For a Randers manifold $(M,F:=\sqrt{g}+β)$, where $g$ is a Riemannian…

The contribution of this paper is two-fold. The first one is to derive a simple formula of the mean curvature form for a hypersurface in the Randers space with a Killing field, by considering the Busemann-Hausdorff measure and Holmes-Thompson measure simultaneously. The second one is to obtain the explicit local expres…

We give a necessary and sufficient condition on a Randers space for the existence of a measure for which Shen's S-curvature vanishes everywhere. Moreover, such a measure coincides with the Busemann-Hausdorff measure up to a constant multiplication.

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

Two Morse-Bott volume forms are diffeomorphic if their cohomology classes are equal.

Natural volume forms defined for pseudo-Finslerian manifolds with specific metrics.

S. Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. We solve the geodesic equation and its pert…

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 vertices. The square-volume gives rise to a characterization of the volume form in the t…

Researchers find highest volumes for isospectral spherical orbifolds and space forms.

Infinite volume moduli spaces of hyperbolic surfaces are redefined with exponential forms.

Study geodesic equation on mixed-volume forms on balanced manifolds, proving existence of solutions.

The paper explores the connection between Poisson-Lie structures and invariant volume forms in Hamiltonian dynamics.

The paper defines unimodularity for coisotropic Poisson spaces and discusses invariant volume forms.

For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the other one is the power of the Atiyah-Bott-Goldman symplectic form. We introduce an h…

Study shows how volume forms on degenerating varieties converge to a non-Archimedean measure.

Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.

For a knot K in S^3 we construct according to Casson--or more precisely taking into account Lin and Heusener's further works--a volume form on the SU(2)-representation space of the group of K. We prove that this volume form is a topological knot invariant and explore some of its properties.

Study intrinsic volume forms on complex hypersurfaces.

Calabi-Yau theorem extended to Vaisman manifolds.

Holomorphic quantum modular forms linked to knot volumes.

Let $M$ be a compact $n$-manifold of $\operatorname{Ric}_M\ge (n-1)H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions: (i) There is $ρ>0$ such that for any $x\in M$, the open $ρ$-ball at $x^…

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

This paper confirms volumes of geodesic balls can identify 4D space forms.

We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a co…

In this paper, we are interested in flat metric structures with conical singularities on surfaces which are obtained by deforming translation surface structures. The moduli space of such flat metric structures can be viewed as some deformation of the moduli space of translation surfaces. Using geodesic triangulations, …

The paper improves the regularity and existence of pseudo Calabi flow.

We prove a stability result for volume forms on fiber bundles with compact base and noncompact fibers. This generalizes the classical results of Moser and Greene--Shiohama, and recent work by the authors.

We calculate finite volumes of moduli spaces of flat surfaces with conical singularities.

We study the flux homomorphism for closed forms of arbitrary degree, with special emphasis on volume forms and on symplectic forms. The volume flux group is an invariant of the underlying manifold, whose non-vanishing implies that the manifold resembles one with a circle action with homologically essential orbits.

We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.

Study shows volume and genus unrelated for hyperbolic fibred knots.

The paper classifies hypersurfaces with constant weighted mean curvature.

A simpler edge-based discretization method without dual volumes.

We prove a $C^{1,1}$ estimate for solutions of a class of fully nonlinear equations introduced by Chen-He. As an application, we prove the $C^{1,1}$ regularity of geodesics in the space of volume forms.

We introduce a flow of Riemannian metrics and positive volume forms over compact oriented manifolds whose formal limit is a shrinking Ricci soliton. The case of a fixed volume form has been considered in our previous work. We still call this new flow the Soliton-Ricci flow. It corresponds to a forward Ricci type flow u…

Smooth symplectic manifolds can be approximated by PL symplectic manifolds.

Given $d\in \mathbb{N}$, $g\in \mathbb{N} \cup\{0\}$, and an integral vector $κ=(k_1,\dots,k_n)$ such that $k_i>-d$ and $k_1+\dots+k_n=d(2g-2)$, let $Ω^d\mathcal{M}_{g,n}(κ)$ denote the moduli space of meromorphic $d$-differentials on Riemann surfaces of genus $g$ whose zeros and poles have orders prescribed by $κ$. We…

We calculate the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.

The study proves a localization theorem and calculates volumes for superspaces.

Rough and Hodge Laplacians eigenvalues approach zero with fixed volume.

Estimates lower bound for simplicial volume of certain manifolds.

We consider a connected smooth $n$-dimensional manifold $M$ endowed with a volume form $Ω$, and we show that an open subset $U$ of $R^n$ of Lebesgue measure $\Vol (U)$ embeds into $M$ by a smooth volume preserving embedding whenever the volume condition $\Vol (U) \le \Vol (M,Ω)$ is met.