Completion Fibres of Infinite Positive Newton Conformal Metrics Near Corners
Abstract
We study the local metric completion of positive singular conformal metrics near a codimension-two corner when the conformal factor is generated by a compact positive measure on exponent space. The Newton support function determines noncritical weighted-ray accessibility, but at the critical power order support geometry alone is insufficient. We identify the missing invariant as a Laplace transform measuring how exponent mass approaches the critical total-degree face. If the maximal total degree exceeds 2, the local completion fibre over the corner is empty. When all total degrees are at most 2, the fibre is empty or a singleton according as the square root of the critical Laplace profile is nonintegrable or integrable at infinity. We give support-identical measures with opposite critical completion behaviour, sharpness examples, monotone truncation phenomena, intrinsic Lipschitz completion coordinates, and a bi-Lipschitz transfer principle for Newton-comparable densities.
Related
- Complete conformal metrics of constant positive scalar curvature with prescribed isolated singularities
- Determination of all diffeomorphism invariant tensor fields on the space of smooth positive densities on a compact manifold with corners
- Rigidity of conformal metrics with constant sixth-order Q-curvature
- Harmonic-Einstein maps and critical metrics for eigenvalues of the conformal Laplacian
- Conformal boundary rigidity for simple Finsler metrics
BibTeX
@misc{abidin2026completion,
title = {{Completion Fibres of Infinite Positive Newton Conformal Metrics Near Corners}},
author = {Muhamad Fahmi Bin Zanal Abidin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27840},
eprint = {2609.27840},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27840},
}
RIS
TY - UNPB
TI - Completion Fibres of Infinite Positive Newton Conformal Metrics Near Corners
AU - Muhamad Fahmi Bin Zanal Abidin
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27840
UR - https://arxiv.org/abs/2609.27840
AB - We study the local metric completion of positive singular conformal metrics near a codimension-two corner when the conformal factor is generated by a compact positive measure on exponent space. The Newton support function determines noncritical weighted-ray accessibility, but at the critical power order support geometry alone is insufficient. We identify the missing invariant as a Laplace transform measuring how exponent mass approaches the critical total-degree face. If the maximal total degree exceeds 2, the local completion fibre over the corner is empty. When all total degrees are at most 2, the fibre is empty or a singleton according as the square root of the critical Laplace profile is nonintegrable or integrable at infinity. We give support-identical measures with opposite critical completion behaviour, sharpness examples, monotone truncation phenomena, intrinsic Lipschitz completion coordinates, and a bi-Lipschitz transfer principle for Newton-comparable densities.
ER -