Yuhao Xue’s research while affiliated with Tsinghua University and other places

In this paper, we investigate the asymptotic behavior of the non-simple systole, which is the length of a shortest non-simple closed geodesic, on a random closed hyperbolic surface on the moduli space $\mathcal{M}_g$ of Riemann surfaces of genus g endowed with the Weil-Petersson measure. We show that as the genus g goes to infinity, the non-simple systole of a generic hyperbolic surface in $\mathcal{M}_g$ behaves exactly like $\log g$.

In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus with respect to the Weil–Petersson measure on the moduli space . We show that as goes to infinity, a generic surface satisfies asymptotically: the separating systole of is approximately there is a half‐collar of width approximately around any separating systolic curve on the length of the shortest separating closed multi‐geodesics on is approximately . As applications, we also discuss the asymptotic behavior of the extremal separating systole, the non‐simple systole, and the expected length of the shortest separating closed multi‐geodesics as goes to infinity.

Let $\mathcal{M}_g$ 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 $g\to \infty$, for a generic surface in $\mathcal{M}_g$, the error term in the Prime Geodesic Theorem is bounded from above by $g\cdot t^{\frac{3}{4}+\epsilon}$, up to a uniform constant multiplication. The expected value of the error term in the Prime Geodesic Theorem over $\mathcal{M}_g$ is also studied. As an application, we show that as $g\to \infty$, on a generic hyperbolic surface in $\mathcal{M}_g$ most geodesics of length significantly less than $\sqrt{g}$ are simple and non-separating, and most geodesics of length significantly greater than $\sqrt{g}$ are not simple, which confirms a conjecture of Lipnowski-Wright. A novel effective upper bound for intersection numbers on $\mathcal{M}_{g,n}$ is also established, when certain indices are large compared to $\sqrt{g+n}$.

Let Mg$\mathcal {M}_g$ be the moduli space of hyperbolic surfaces of genus g endowed with the Weil–Petersson metric. In this paper, we show that for any ϵ>0$\epsilon >0$, as genus g goes to infinity, a generic surface X∈Mg$X\in \mathcal {M}_g$ satisfies that the first eigenvalue λ1(X)>316-ϵ$\lambda _1(X)>\frac{3}{16}-\epsilon$. As an application, we also show that a generic surface X∈Mg$X\in \mathcal {M}_g$ satisfies that the diameter diam(X)<(4+ϵ)ln(g)${{\,\mathrm{diam}\,}}(X)<(4+\epsilon )\ln (g)$ for large genus.

In this note we show that the expected value of the separating systole of a random surface of genus g with respect to Weil–Petersson volume behaves like $2\log g$ as the genus goes to infinity. This is in strong contrast to the behavior of the expected value of the systole which, by results of Mirzakhani and Petri, is independent of genus.

In this article we study the first eigenvalues of closed Riemann surfaces for large genus. We show that for every closed Riemann surface $X_g$ of genus g $(g\geq 2)$, the first eigenvalue of $X_g$ is greater than $\frac{\mathcal{L}_1(X_g)}{g^2}$ up to a uniform positive constant multiplication. Where $\mathcal{L}_1(X_g)$ is the shortest length of multi closed curves separating $X_g$. Moreover,we also show that this new lower bound is optimal as $g \to \infty$.

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 eigenvalue $\lambda_1(X)>\frac{3}{16}-\epsilon$.

In this note we show that the expected value of the separating systole of a random surface of genus g with respect to Weil-Petersson volume behaves like $2\log g$ as the genus goes to infinity.

In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus g with respect to the Weil-Petersson measure on the moduli space $\mathcal{M}_g$. We show that as g goes to infinity, a generic surface $X\in \mathcal{M}_g$ satisfies asymptotically: (1) the separating systole of X is about $2\log g$; (2) there is a half-collar of width about $\frac{\log g}{2}$ around a separating systolic curve of X; (3) the length of shortest separating closed multi-geodesics of X is about $2\log g$. As applications, we also discuss the asymptotic behavior of the extremal separating systole and the expectation value of lengths of shortest separating closed multi-geodesics as g goes to infinity.