arXiv subjects
Publications and source records attributed to Qiliang Luo.
Given any closed hyperbolic n-manifold M ($n\ge 3$), we show that for a fixed $κ>\frac{3}{n-2}$, a random closed geodesic of length close to r has the collar of width $r^{-κ}$ 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.
We prove that every closed hyperbolic surface admits a sequence of finite covers with asymptotically optimal diameters.
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.
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.