Related papers: Logarithmic systolic growth for hyperbolic surface…

Let N be a manifold (with boundary) of dimension at least 3, such that its interior admits a hyperbolic metric of finite volume. We discuss the possible limits arising from sequences of relative fundamental cycles approximating the…

In this article we study the differences of two consecutive eigenvalues $\lambda_{i}-\lambda_{i-1}$ up to $i=2g-2$ for the Laplacian on hyperbolic surfaces of genus $g$, and show that the supremum of such spectral gaps over the moduli space…

It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal…

We study the long-time behaviour of the focusing cubic NLS on $\R$ in the Sobolev norms $H^s$ for $0 < s < 1$. We obtain polynomial growth-type upper bounds on the $H^s$ norms, and also limit any orbital $H^s$ instability of the ground…

A basic feature of Teichm\"uller theory of Riemann surfaces is the interplay of two dimensional hyperbolic geometry, the behavior of geodesic-length functions and Weil-Petersson geometry. Let $\mathcal{T}_g$ $(g\geq 2)$ be the Teichm\"uller…

On a hyperbolic 3-manifold of finite volume, we prove that if the initial metric is sufficiently close to the hyperbolic metric $h_0$, then the normalized Ricci-DeTurck flow exists for all time and converges exponentially fast to $h_0$ in a…

In this article, we provide bounds on systoles associated to a holomorphic $1$-form $\omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on…

We show existence of constant mean curvature 1 surfaces in both hyperbolic 3-space and de Sitter 3-space with two complete embedded ends and any positive genus up to genus twenty. We also find another such family of surfaces in de Sitter…

We prove the existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds provided there are barriers.

We show that for every $n\geq 2$ and any $\epsilon>0$ there exists a compact hyperbolic $n$-manifold with a closed geodesic of length less than $\epsilon$. When $\epsilon$ is sufficiently small these manifolds are non-arithmetic, and they…

We obtain polynomial bounds on the growth in time of Sobolev norm of solutions to the cubic defocusing nonlinear Schrodinger equation on two dimensional product space. We also give the angular improved bilinear Strichartz estimates for…

We gave an alternative short proof on the finite generation of holomorphic functions with polynomial growth on Riemann surfaces with nonnegative curvature. The first proof was due to Li and Tam.

We show that for $k>1$ the number of genus $k$ minimal Lagrangians with area at most $A$ in a product of hyperbolic surfaces grows on the order of $A^{6(k-1)}$, with an explicit leading constant given in terms of the Mirzakhani function. We…

We present a proof of a conjecture proposed by V. Delecroix, E. Goujard, P. Zograf, and A. Zorich, which describes the large genus asymptotic behaviours of the ratio of frequencies of separating over nonseparating simple closed geodesics on…

We present the Laplace operator associated to a hyperbolic surface $\Gamma\setminus\mathbb{H}$ and a unitary representation of the fundamental group $\Gamma$, extending the previous definition for hyperbolic surfaces of finite area to those…

We consider circles of common centre and increasing radius on a compact hyperbolic surface and, more generally, on its unit tangent bundle. We establish a precise asymptotics for their rate of equidistribution. Our result holds for…

We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity properties. Using these properties, we…

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…

We show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in $\mathbf{R}^4$ has intrinsic cubic volume growth, provided the parametric elliptic integral is $C^2$-close to the area functional. We also obtain an…

In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which…