AI 中文总结
该研究在度量假设下探讨曲面孤立奇点的共形结构延拓，证明M-正则孤立奇点是共形点状的，给出亚解析曲面奇点的推论及说明M-正则两假设独立的C^∞反例。
AI 中文摘要
我们在非解析而是度量的假设下，研究作为Rⁿ子流形得到的黎曼曲面的共形结构在孤立奇点处的延拓问题。1989年的主要结果表明：当孤立奇点是**M-正则**时，它是**共形点状的**（共形于一个穿孔圆盘）；这里的M-正则指的是（去掉该点的曲面，该点）满足Whitney条件，且球切片S(0,r)∩E的长度随r的衰减至多为线性，该条件被称为“度量衰减”。这一结果通过环模（极值长度）论证证明，将经典平面技术推广到Rⁿ的子流形。文中处理了两类例子：由单条曲线生成的旋转曲面，以及亚解析曲面；对亚解析曲面可直接从Hironaka结构理论验证M-正则性，由此得到推论：每个可定向亚解析曲面的孤立奇点都是共形点状的。还有一个原创结果是一个C^∞反例，表明仅严格Whitney条件本身**不**隐含线性长度界，即M-正则性定义中的两个假设是相互独立的。
英文摘要
We study the extension of the conformal structure of a Riemann surface, obtained as a submanifold of $\mathbf R^n$, across an isolated singular point, under hypotheses that are metric rather than analytic. The main result (1989) is that an isolated singularity is $\textbf{conformally point-like}$ (conformal to a punctured disc) whenever it is $\textbf{$M$-regular}$: the pair (surface minus the point, the point) satisfies a Whitney condition, and the length of the spherical slice $S(0,r)\cap E$ decreases at most linearly in $r$, a condition described as ``metric decay''. This is proved via a modulus-of-rings (extremal-length) argument, generalizing the classical planar technique to submanifolds of $\mathbf R^n$. Two classes of examples are treated: surfaces of revolution generated by a single curve, and subanalytic surfaces, for which $M$-regularity is verified directly from Hironaka's structure theory, giving as a corollary that every isolated singularity of an orientable subanalytic surface is conformally point-like. A further original result is a $C^\infty$ counterexample showing that a strict Whitney condition alone does $\textbf{not}$ imply the linear length bound: the two hypotheses in the definition of $M$-regularity are independent.
Comments2026 Preprint 1.7 (441), Dipartimento di Matematica, Sezione di Geometria e Algebra, Universita di Pisa, April 1989
