Yuhao Xue's research works | Tsinghua University, Beijing (TH) and other places

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Figure 1. A figure-eight closed geodesic α in the pair of pants P (x,...

Figure 2. Upper left: X \ P (x, y, z) ∼ = S 1,3 is connected. Upper...

Non-simple systoles on random hyperbolic surfaces for large genus

Preprint

Full-text available

Aug 2023

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-sim...

Large genus asymptotics for lengths of separating closed geodesics on random surfaces

Article

Jan 2023

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...

Prime geodesic theorem and closed geodesics for large genus

Preprint

Full-text available

Sep 2022

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 con...

Optimal lower bounds for first eigenvalues of Riemann surfaces for large genus

Article

Aug 2022

Random hyperbolic surfaces of large genus have first eigenvalues greater than $$\frac{3}{16}-\epsilon

Article

Full-text available

Apr 2022

Let Mg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}_g$$\end{document} be the moduli space of hyperbolic surfaces of genus g endowed with the Weil–Peterss...

THE SIMPLE SEPARATING SYSTOLE FOR HYPERBOLIC SURFACES OF LARGE GENUS

Article

May 2021

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.

Optimal lower bounds for first eigenvalues of Riemann surfaces for large genus

Preprint

Full-text available

Mar 2021

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...

Figure 3. Y 1 ∼ = Y 2 ∼ = Y 3 ∼ = S 0,3 , but they are in different orbits

Figure 7. an example for η in P In the rectangle with right angle...

Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac{3}{16}-\epsilon

Preprint

Full-text available

Feb 2021

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

Simple separating systole of hyperbolic surfaces for large genus

Preprint

Full-text available

Dec 2020

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.

Large genus asymptotics for lengths of separating closed geodesics on random surfaces

Preprint

Full-text available

Sep 2020

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...

Small eigenvalues of closed Riemann surfaces for large genus

Preprint

Full-text available

Sep 2018

In this article we study the asymptotic behavior of small eigenvalues of hyperbolic surfaces for large genus. We show that for any positive integer k k , as the genus g g goes to infinity, the minimum of k k -th eigenvalues of hyperbolic surfaces over any thick part of moduli space of Riemann surfaces of genus g g is uniformly comparable to 1 g 2 \...