A problem for isolated singularities of surfaces
Abstract
Let E⊂Rn be a set of dimension $2$ with an isolated singular point $0$, such that E−{0} is an orientable 𝒞2 submanifold of Rn 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 $0$ M−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)∩ E is bounded above by a linear function of r for small r (metric decay). The main result (1989) is that an M-regular singularity is conformally point−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 ℝn. Two classes of surfaces are shown to be conformally point-like: all surfaces of revolution generated by a single C1 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)∩ 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∞ counterexample showing that a strict Whitney condition alone does not imply the linear length bound: the two hypotheses defining M-regularity are independent.
Related
- The Yau Geometric Conjecture for Isolated Hypersurface Singularities
- Isolated symplectic singularities admitting symplectic resolutions
- F-nilpotence and Hodge filtrations beyond isolated singularities
- The Kemper--Malle conjecture and isolated linear quotient singularities
- Motivic local density of isolated singularities of algebraic varieties
BibTeX
@misc{mosca2026problem,
title = {{A problem for isolated singularities of surfaces}},
author = {Flavio Mosca},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.19452},
eprint = {2608.19452},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19452},
}
RIS
TY - UNPB
TI - A problem for isolated singularities of surfaces
AU - Flavio Mosca
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2608.19452
UR - https://arxiv.org/abs/2608.19452
AB - Let E⊂R^n be a set of dimension $2$ with an isolated singular point $0$, such that E−{0} is an orientable 𝒞^2 submanifold of 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 $0$ M−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)∩ E is bounded above by a linear function of r for small r (metric decay). The main result (1989) is that an M-regular singularity is conformally point−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 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)∩ 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^∞ counterexample showing that a strict Whitney condition alone does not imply the linear length bound: the two hypotheses defining M-regularity are independent.
ER -