Year: 2026
Type: preprint
Abstract: Let $E\subset\mathbf R^n$ be a set of dimension $2$ with an isolated singular point $0$, such that $E-\{0\}$ is an orientable $\mathcal C^2$ submanifold of $\mathbf R^n$ of dimension $2$, and hence a Riemann surface with the conformal structure induced by the ambient metric. We study the extension of its conformal structure across $0$, under hypotheses that are metric rather than analytic. Denoting by $S(0,r)$ the sphere of radius $r$ centred at $0$, we call the singularity $0$ $\mathit{M\text{-}regular}$ if (i) the pair $(E-\{0\},\{0\})$ satisfies a Whitney condition (the radial direction becomes asymptotically tangent to $E-\{0\}$ at $0$) and (ii) the length of the spherical slice $S(0,r)\cap E$ is bounded above by a linear function of $r$ for small $r$ ($\mathit{metric\ decay}$). The main result (1989) is that an $M$-regular singularity is $\mathit{conformally\ point\text{-}like}$, that is, a punctured neighbourhood of $0$ in $E$ is conformal to the punctured unit disc $Δ^*$, so that the conformal structure extends across $0$. The proof uses the modulus of rings (extremal length), which generalizes the classical planar technique to $2$-dimensional submanifolds of $\mathbb{R}^n$. Two classes of surfaces are shown to be conformally point-like: all surfaces of revolution generated by a single $C^1$ curve, and orientable subanalytic sets of dimension $2$ with isolated singularities, for which $M$-regularity is proven directly from Hironaka's structure theory. In this case the metric decay condition follows from a direct estimate of the volume of the slices $S(0,r)\cap E$ (Lemma 7.5), based on Hironaka's Whitney theorem and Stasica's decomposition into $L$-analytic pieces. A further original result is a $C^\infty$ counterexample showing that a strict Whitney condition alone does $\mathit{not}$ imply the linear length bound: the two hypotheses defining $M$-regularity are independent.
Source: arXiv (Cornell University)
Author: Mosca, Flavio
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203857105