Metric Completion and Boundary Geometry of Multi-Weighted Conformal Metrics on Manifolds with Corners

Abstract

We study metric completions of conformally singular Riemannian metrics on the interior of a compact manifold with corners. Given boundary-defining functions \(\rho_1,\ldots,\rho_N\) and nonnegative weights \(\alpha_1,\ldots,\alpha_N\), we compare the product metric \(g_\Pi=(\prod_i\rho_i^{-\alpha_i})g_0\) with the sum metric \(g_\Sigma=(\sum_i\rho_i^{-\alpha_i})g_0\). For a boundary point \(p\), let \(I(p)\) be its active index set, \(A(p)=\sum_{i\in I(p)}\alpha_i\), and \(M(p)=\max_{i\in I(p)}\alpha_i\). 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.