Huber Theorem revisited in dimensions 2 and 4

Paul Laurain; Dorian Martino

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arXiv · 2502.05541

Huber Theorem revisited in dimensions 2 and 4

Paul Laurain·Dorian Martino

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Abstract

We study the second Huber theorem in dimensions 2 and 4. In dimension 2, we prove a new version assuming that the Gauss curvature lies in a negative Sobolev space using Coulomb frames. In dimension $4$, given a metric having a pointwise singularity with $L^p$-bounds on the Bach tensor, we construct a conformal metric which is regular across the singularity. To do so, we introduce another Coulomb-type condition, similar to the case of Yang--Mills connections. This enables us to obtain a conformal metric satisfying an $\varepsilon$-regularity property. We obtain a generalization of the two-dimensional case that can be applied to study the singularities of Bach-flat metrics and immersions with second fundamental forms in $W^{2,\frac{4}{3}+\varepsilon}$.

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BibTeX RIS

Paul Laurain, Dorian Martino. 2025-02-08. Huber Theorem revisited in dimensions 2 and 4. https://arxiv.org/abs/2502.05541

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