Related papers: Explicit upper bound for the Weil-Petersson volume…

In this article we study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any positive integer $k$, as the genus $g$ goes to infinity, the smallest $k$-th eigenvalue of Riemann surfaces in…

For several embedded surfaces with zero self-intersection number in 4-manifolds, we show that an adjunction-type genus bound holds for at least one of the surfaces under certain conditions. For example, we derive certain adjunction…

We compute the CM volume, that is the degree of the descended CM line bundle, of the Fano K-moduli space of Quartic del Pezzo in any dimension, and of the K-moduli space of the log Fano hyperplane arrangements of dimension one and two.…

For any closed orientable 3-manifold, there is a volume function defined on the space of all Seifert representations of the fundamental group. The maximum absolute value of this function agrees with the Seifert volume of the manifold due to…

Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n$, for $3 \leq n \leq 7$, and non-negative Ricci curvature. Let $g = \phi^2 g_0$ be a metric in the conformal class of $g_0$. We show that there exists a smooth closed embedded…

We show that the introduction of triangulations with variable connectivity and fluctuating egde-lengths (Random Regge Triangulations) allows for a relatively simple and direct analyisis of the modular properties of 2 dimensional simplicial…

This article follow the article {http://hal.archives-ouvertes.fr/hal-00361030/fr/} in which the author characterize the fact of being of finite volume for a convex projective surface. We show here that the moduli space…

We prove a Gauss-Bonnet theorem for (finite coverings of) moduli spaces of Riemann surfaces endowed with the McMullen metric. The proof uses properties of an exhaustion of moduli spaces by compact submanifolds with corners and the…

Triangulated surfaces are compact Riemann surfaces equipped with a conformal triangulation by equilateral triangles. In 2004, Brooks and Makover asked how triangulated surfaces are distributed in the moduli space of Riemann surfaces as the…

In this paper, we study spectral gaps of closed hyperbolic surfaces for large genus. We show that for any fixed $k\geq 1$, as the genus goes to infinity, the maximum of $\lambda_k-\lambda_{k-1}$ over any thick part of the moduli space of…

The regularity of limit spaces of Riemannian manifolds with L^p curvature bounds, $p > n/2$, is investigated under no apriori non-collapsing assumption. A regular subset, defined by a local volume growth condition for a limit measure, is…

Let $N_{g,n}$ denote the nonorientable surface of genus $g$ with $n$ boundary components and $M(N_{g,n})$ its mapping class group. We obtain an explicit finite presentation of $M(N_{g,n})$ for $n=0,1$ and all $g$ such that $g+n>3$.

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient over the…

Let $M_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. In this paper, we show that for any $\epsilon>0$, as genus $g$ goes to infinity, a generic surface $X\in M_g$ satisfies that the first…

Let $M$ denote a compact, connected Riemannian manifold of dimension $n\in{\mathbb N}$. We assume that $ M$ has a smooth and connected boundary. Denote by $g$ and ${\rm d}v_g$ respectively, the Riemannian metric on $M$ and the associated…

In this article, we derive estimates of Teichm\"uller modular forms, and associated invariants. Let $\mathcal{M}_{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, and let $\overline{M}_{g}$ be the…

Let $(M^n,g)$ be a complete Riemannian manifold which is not isometric to $\mathbb{R}^n$, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set $\mathcal{G}\subset…

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…

The renormalized volume is a smooth function associating to every convex co-compact hyperbolic $3$-manifold $M$ a real number. When the boundary of $M$ is incompressible, the renormalized volume is always positive, otherwise there are…

In this paper, we are interested in the problem of scattering by strictly convex obstacles in the plane. We provide an upper bound for the number $N(r,\gamma)$ of resonances in the box $\{r \le \Re(\lambda) \le r + 1$; $\Im(\lambda) \ge -…