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Qiliang Luo

Publications and source records attributed to Qiliang Luo.

4 recordsLinked to original sources
arXiv2026

Simple geodesics in closed hyperbolic manifolds

Given any closed hyperbolic n-manifold M ($n\ge 3$), we show that for a fixed $\kappa>\frac{3}{n-2}$, a random closed geodesic of length close to r has the collar of width $r^{-\kappa}$ when r is large enough. In particular, most closed geodesics of length close to r are simple. This theorem positively answers Problem 3.16 from Kirby's list which asks whether every closed hyperbolic 3-manifold contains infinitely many simple closed geodesics.

math.GT↗
arXiv2026

The Effective Ehrenpreis Conjecture

Let $M$ and $N$ be two closed hyperbolic Riemann surfaces. The Ehrenpreis Conjecture (proved by Kahn-Markovic) asserts that for any $ε>0$ there are finite covers $M_ε\to M$, and $N_ε\to N$, such that the Teichmuller distance (in the suitable moduli space) between $M_ε$ and $N_ε$ is less than $ε$. It is natural to ask how large the degrees of these coverings need to be to achieve that the distance between $M_ε$ and $N_ε$ is less than $ε$. In this paper we show that there exists a constant $k>0$, depending only on $M$ and $N$, so that the covers $M_ε\to M$, and $N_ε\to N$, can be chosen to have the degrees less than $ε^{-k}$. We show that this bound is optimal by considering the case when $M$ and $N$ are arithmetic Riemann surfaces with the same invariant trace field which are not commensurable to each other.

math.GT↗
arXiv2025

Unique extremality of affine maps on plane domains

We prove that affine maps are uniquely extremal quasiconformal maps on the complement of a well distribute set in the complex plane answering a conjecture from \cite{markovic}. We construct the required Reich sequence using Bergman projections, and meromorphic partitions of unity.

math.CV↗