arXiv subjects

Yunpeng Meng

Publications and source records attributed to Yunpeng Meng.

4 recordsLinked to original sources

arXiv2026

On the Volume Conjectures for Dehn-filled $3$-manifolds along fundamental shadow link complements with big volume

We prove the Volume Conjecture and the Asymptotic Expansion Conjecture for closed oriented hyperbolic $3$-manifolds obtained by integral Dehn-fillings along a fundamental shadow link complement, provided that their volumes are sufficiently close to that of the complement. More generally, we show that the conjectured asymptotic expansions hold for pairs obtained by change-of-pair operations on a fundamental shadow link complement whenever the volume of the corresponding hyperbolic cone manifold is sufficiently close to that of the complement. A key ingredient in the proof is the introduction of extended quantum $6j$-symbols and a Weak Reflection Symmetry satisfied by them. The Weak Reflection Symmetry is established by relating the extended quantum $6j$-symbols to the $b$-$6j$ symbols through a real Gevrey-$1$ estimate and a truncated Borel-Laplace transform.

math.GT↗

arXiv2026

Asymptotics of complex $b$-$6j$ symbols

We study the $b$-$6j$ symbols -- an analytic extension of the $6j$-symbols for the principal series of the modular double of $\mathrm U_q\mathfrak{sl}(2;\mathbb R)$ -- with complex index $b,$ refereed to as the \emph{complex $b$-$6j$ symbols}. Then we relate their asymptotics, when the six parameters scale according to the dihedral angles of a hyperideal hyperbolic tetrahedron, to the volume and the determinant of the Gram matrix of the tetrahedron. In the case $\arg b=\pm \fracπ{4},$ we believe that this work is closely related to the complex Liouville string\,\cite{CEMR}.

math-ph↗

arXiv2025

Hyperbolization and geometric decomposition of a class of 3-manifolds

Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric triangulation into hyper-ideal hyperbolic tetrahedra. So far, this conjecture had only been proven for a few special 3-manifolds. In this article, we confirm this conjecture for a class of 3-manifolds. To be precise, let $M$ be an oriented compact 3-manifold with boundary, no component of which is a 2-sphere, and $\mathcal{T}$ is an ideal triangulation of $M$. If $\mathcal{T}$ satisfies properly gluing condition, and the valence is at least 6 at each ideal edge and 11 at each hyper-ideal edge, then $M$ admits an unique complete hyperbolic metric with totally geodesic boundary, so that $\mathcal{T}$ is isotopic to a geometric ideal triangulation of $M$. We use analytical tools such as combinatorial Ricci flow (CRF, abbr.) to derive the conclusions. There are intrinsic difficulties in dealing with CRF. First, the CRF may collapse in a finite time, second, most of the smooth curvature flow methods are no longer applicable since there is no local coordinates in $\mathcal{T}$, and third, the evolution of CRF is affected by certain combinatorial obstacles in addition to topology. To this end, we introduce the ideas as ``extending CRF", ``tetrahedral comparison principles", and ``control CRF with edge valence" to solve the above difficulties. In addition, the presence of torus boundary adds substantial difficulties in this article, which we have solved by introducing the properly gluing conditions on $\mathcal{T}$ and reducing the ECRF to a flow relatively easy to handle.

math.GT↗

arXiv2024

On the Volume Conjecture for hyperbolic Dehn-filled 3-manifolds along the twist knots

For a twist knot $\mathcal{K}_{p'}$, let $M$ be the closed $3$-manifold obtained by doing $(p, q)$ Dehn-filling along $\mathcal{K}_{p'}$. In this article, we prove that Chen-Yang's volume conjecture holds for sufficiently large $|p| + |q|$ and $|p'|$ for $M$. In the proof, we construct a new ideal triangulation of the Whitehead link complement which is different from Thurston's triangulation. Our triangulation has led to some new discoveries regarding symmetry, including insights into ``sister manifolds'' as introduced by Hodgson, Meyerhoff, and Weeks.

math.GT↗