带角流形上多权共形度量的度量完备与边界几何
Metric Completion and Boundary Geometry of Multi-Weighted Conformal Metrics on Manifolds with Corners
AI总结：
该研究针对带角紧流形内部的共形奇异黎曼度量，分析乘积与和两类多权共形度量的完备条件、边界几何及豪斯多夫维数，揭示两类模型的边界可达性差异。
AI中文摘要：
我们研究带角的紧流形内部共形奇异黎曼度量的度量完备性。给定边界定义函数ρ₁,…,ρ_N及非负权重α₁,…,α_N，我们比较乘积度量g_Π=(∏ᵢρᵢ^(-αᵢ))g₀与和度量g_Σ=(∑ᵢρᵢ^(-αᵢ))g₀。对边界点p，设I(p)为其有效指标集，A(p)=∑ᵢ∈I(p)αᵢ，M(p)=maxᵢ∈I(p)αᵢ。我们证明：对于乘积度量，p处于有限度量距离上当且仅当A(p)<2；对于和度量，当且仅当M(p)<2。在每个可达边界点上，有效法向坍缩为单个完备点，给出完备性的典范拓扑描述。相互作用规律产生不同的边界可达性行为：乘积权重可使得即使其组成超曲面各自可达，更深的面也不可达，而和模型由最大有效权重决定。我们还确定了可达开面上的诱导几何：局部上，乘积情形下边界度量是指数为1−A_I/2的雪花形，和情形下是指数为1−M_I/2的雪花形，对应豪斯多夫维数分别为m/(1−A_I/2)和m/(1−M_I/2)。
英文摘要：
We study metric completions of conformally singular Riemannian metrics on the interior of a compact manifold with corners. Given boundary-defining functions \(ρ_1,\ldots,ρ_N\) and nonnegative weights \(α_1,\ldots,α_N\), we compare the product metric \(g_Π=(\prod_iρ_i^{-α_i})g_0\) with the sum metric \(g_Σ=(\sum_iρ_i^{-α_i})g_0\). For a boundary point \(p\), let \(I(p)\) be its active index set, \(A(p)=\sum_{i\in I(p)}α_i\), and \(M(p)=\max_{i\in I(p)}α_i\). We prove that \(p\) lies at finite metric distance precisely when \(A(p)<2\) for the product metric and \(M(p)<2\) for the sum metric. Over every accessible boundary point, the active normal directions collapse to a single completion point, yielding a canonical topological description of the completion. The interaction laws produce different boundary-incidence behavior: product weights can make a deeper face inaccessible even when its constituent hypersurfaces are individually accessible, whereas the sum model is governed by the largest active weight. We also determine the induced geometry on accessible open faces: locally the boundary metric is a snowflake with exponent \(1-A_I/2\) in the product case and \(1-M_I/2\) in the sum case, giving Hausdorff dimensions \(m/(1-A_I/2)\) and \(m/(1-M_I/2)\), respectively.
