角处奇异共形度量的牛顿支撑函数与度量完备化
Newton Support Functions and Metric Completion of Singular Conformal Metrics at Corners
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中文总结 AI 辅助
该研究针对余维二角附近的奇异共形度量，利用牛顿支撑函数证明射影可达性准则，推导有限度量距离角的判定条件，得到边界度量性质与豪斯多夫维数公式，还涉及有理指数数据的牛顿分解实现。
中文摘要 AI 辅助
我们研究余维二角附近的奇异共形度量\boldsymbol{g=Fg_0}，其中\boldsymbol{F}是单项式奇点的有限正和。竞争阶次由齐次牛顿支撑函数\boldsymbol{\textit{H}(p,q)=\text{max}_j(pa_j+qb_j)}编码。我们证明了尖锐射影可达性准则\boldsymbol{\textit{H}(p,q)<2\text{min}\{p,q\}}，并推导出仅当\boldsymbol{\text{max}_j(a_j+b_j)<2}时，角才处于有限度量距离内。在每个可达角处，固定角点上的所有法向逼近确定一个典范完备点，诱导的边界度量与光滑角度量的雪花形是局部双李普希茨等价的，由此得到显式豪斯多夫维数公式。对于有理指数数据，牛顿分解也可通过兼容的有根修改和单态修改实现。度量结论本身仅要求正性、有界系数及一致椭圆性。
英文摘要
We study singular conformal metrics \(g=Fg_0\) near codimension-two corners, where \(F\) is a finite positive sum of monomial singularities. The competing orders are encoded by the homogeneous Newton support function \(\mathcal H(p,q)=\max_j(pa_j+qb_j)\). We prove the sharp projective accessibility criterion \(\mathcal H(p,q)<2\min\{p,q\}\), and deduce that the corner lies at finite metric distance exactly when \(\max_j(a_j+b_j)<2\). At every accessible corner, all normal approaches over a fixed corner point determine one canonical completion point. The induced boundary metric is locally bi-Lipschitz to a snowflake of the smooth corner metric, yielding an explicit Hausdorff-dimension formula. For rational exponent data, the Newton decomposition is also realized by compatible rooted and monoidal modifications. The metric conclusions themselves require only positivity, bounded coefficients, and uniform ellipticity.
