Muhamad Fahmi bin Zanal Abidin

arXiv2026

Newton Support Functions and Metric Completion of Singular Conformal Metrics at Corners

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.

math.DG↗

arXiv2026

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

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.

math.DG↗