arXiv subjects

Yaosong Yang

Publications and source records attributed to Yaosong Yang.

4 recordsLinked to original sources

arXiv2026

Decomposition and characterization of VMO via vanishing Carleson measures

We establish two equivalent characterizations of $\mathrm{VMO}$ in terms of vanishing Carleson measures. First, we show that any $\mathrm{VMO}$ function admits a decomposition into a continuous boundary term and an integral operator associated with a vanishing Carleson measure. Second, motivated by Varopoulos's work on the $\bar{\partial}$-equation, we characterize $\mathrm{VMO}$ via the boundary values of smooth functions whose gradients induce vanishing Carleson measures. As a consequence, we recover the known representation \[ \mathrm{VMO}=\mathrm{VLO}-\mathrm{VLO}, \] thereby providing a new perspective on this decomposition.

math.CV↗

arXiv2025

The growth rate on the volume of $\mathcal{M}_g^{<L(g)}$

Let $\mathcal{M}_g$ be the moduli space of hyperbolic surfaces of genus g endowed with the Weil-Petersson metric. In this paper, we introduce a function $L(g)$ of genus $g$ and call the geodesics whose length less than $L(g)$ short geodesics. We compute the growth rate on the volume of the subset of hyperbolic surfaces with short geodesics. In particular, when $g$ approaches infinity, if $L(g)$ also approaches infinity, then the volume of surfaces characterized by short geodesics is equal to $V_g$ almost surely.

math.GT↗

arXiv2025

On Loewner energy and curve composition

The composition $γ\circ η$ of Jordan curves $γ$ and $η$ in universal Teichmüller space is defined through the composition $h_γ\circ h_η$ of their conformal weldings. We show that whenever $γ$ and $η$ have finite Loewner energy $I^L$, the energy of their composition satisfies $$I^L(γ\circ η) \lesssim_K I^L(γ) + I^L(η),$$ with an explicit constant in terms of the quasiconformal $K$ of $γ$ and $η$. We also study the asymptotic growth rate of the Loewner energy under $n$ self-compositions $γ^n := γ\circ \cdots \circ γ$, showing $$\limsup_{n \rightarrow \infty} \frac{1}{n}\log I^L(γ^n) \lesssim_K 1,$$ again with explicit constant. Our approach is to define a new conformally-covariant rooted welding functional $W_h(y)$, and show $W_h(y) \asymp_K I^L(γ)$ when $h$ is a welding of $γ$ and $y$ is any root (a point in the domain of $h$). In the course of our arguments we also give several new expressions for the Loewner energy, including generalized formulas in terms of the Riemann maps $f$ and $g$ for $γ$ which hold irrespective of the placement of $γ$ on the Riemann sphere, the normalization of $f$ and $g$, and what disks $D, \overline{D}^c \subset \hatC$ serve as domains. An additional corollary is that $I^L(γ)$ is bounded above by a constant only depending on the Weil--Petersson distance from $γ$ to the circle.

math.CV↗

arXiv2024

Conformal weldings in the Loewner equation and Weil--Petersson quasislit-disks

A simple arc $Γ= γ(0, T]$, growing into the unit disk $\mathbb D$ from its boundary, generates a driving term $ξ$ and a conformal welding $ϕ$ through the Loewner differential equation. When $Γ$ is the slit of a Weil--Petersson quasislit-disk $\mathbb D\setminusΓ$, the Loewner transform and its inverse $Γ\leftrightarrow ξ$ have been well understood due to Y. Wang's work. We investigate the maps $Γ\leftrightarrow ϕ$ in this case, giving a description of $Γ$ in terms of $ϕ$.

math.CV↗