Related papers: Simplicial volume of Hilbert modular varieties

The moduli spaces of compact and connected Riemann surfaces has been a central topic in modern mathematics in recent years. Thus their homological dimensions become important invariants. Motivated by the emergence mathematical counterparts…

It was proved by Graham and Witten in 1999 that conformal invariants of submanifolds can be obtained via volume renormalization of minimal surfaces in conformally compact Einstein manifolds. The conformal invariant of a submanifold $\Sigma$…

We prove the Proportionality Principle for the Lipschitz simplicial volume without any restriction on curvature. Our argument is based on the construction of a suitable \emph{pseudostraightening} for simplices.

Given a closed Riemannian manifold of dimension $n$ and a Morse-Smale function, there are finitely many $n$-part broken trajectories of the negative gradient flow. We show that if the manifold admits a hyperbolic metric, then the number of…

Volumes of moduli spaces of hyperbolic cone surfaces were previously defined and computed when the angles of the cone singularities are at most 2pi. We propose a general definition of these volumes without restriction on the angles. This…

We study lower semi-continuity properties of the volume, i.e., the surface area, of a closed Lagrangian manifold with respect to the Hofer- and $\gamma$-distance on a class of monotone Lagrangian submanifolds Hamiltonian isotopic to each…

We consider the moduli space of the extremal K\"ahler metrics on compact manifolds. We show that under the conditions of two-sided total volume bounds, $L^{n\over2}$-norm bounds on $\Riem$, and Sobolev constant bounds, this Moduli space can…

It is proved that the simplex is a strict local minimum for the volume product, P(K)=min(vol(K) vol(K^z)), K^z is the polar body of K with respect to z, the minimum is taken over z in the interior of K, in the Banach-Mazur space of…

Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that…

A celebrated result of Mirzakhani states that, if $(S,m)$ is a finite area \emph{orientable} hyperbolic surface, then the number of simple closed geodesics of length less than $L$ on $(S,m)$ is asymptotically equivalent to a positive…

In this paper, we prove the existence of the free boundary minimal hypersurface of least area in compact manifolds with boundary. Such hypersurface can be viewed as the ground state of the volume spectrum introduced by Gromov. Moreover, we…

In this paper we will explore fundamental constraints on the evolution of certain symplectic subvolumes possessed by any Hamiltonian phase space. This research has direct application to optimal control and control of conservative mechanical…

We formulate a conjecture that arithmetic locally symmetric manifolds have simple homotopy type, and prove it for the non-compact case. More precisely, we show that, for any symmetric space S of non-compact type without Euclidean de Rham…

We show that for $n \neq 1,4$ the simplicial volume of an inward tame triangulable open $n$-manifold $M$ with amenable fundamental group at infinity at each end is finite; moreover, we show that if also $\pi_1(M)$ is amenable, then the…

The Shadowing Principle of Manin has proved a valuable tool for addressing questions of quantitative topology raised by Gromov in the late 1900s. The principle informally provides a way for bounded algebraic maps between differential graded…

We consider the problem of estimating the volume of a compact domain in a Euclidean space based on a uniform sample from the domain. We assume the domain has a boundary with positive reach. We propose a data splitting approach to correct…

In a recent work [2] with Datta, we introduced the mu vector (with respect to a given field) of simplicial complexes and used it to study tightness and lower bounds. In this paper, we modify the definition of mu vectors. With the new…

We extend results by Mirzakhani in [Mir07] to moduli spaces of Hurwitz covers. In particular we obtain equations relating Weil-Petersson volumes of moduli spaces of Hurwitz covers, Hurwitz numbers and certain Hurwitz cycles on…

Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces…

For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{C}^{n}$ with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point…