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Complete conformal metrics of constant positive scalar curvature with prescribed isolated singularities
Abstract
Let (Mn,g) be a smooth closed connected Riemannian manifold of dimension n≥3 with positive Yamabe constant. For any prescribed finite set Λ⊂ M containing at least two points, we construct a complete metric on M∖Λ, conformal to g, with scalar curvature n(n−1). At each singularity, the local geometric term in the reduced action comes from the first nonzero homogeneous metric term of degree at most (n−2)/2 in finite conformal normal coordinates, or from the mass of the Green function metric when all such terms vanish. We completely solve Schoen's conjecture about singular Yamabe problem.
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BibTeX
@misc{chen2026complete,
title = {{Complete conformal metrics of constant positive scalar curvature with prescribed isolated singularities}},
author = {Hongyu Chen and Li Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31058},
eprint = {2610.31058},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31058},
}
RIS
TY - UNPB
TI - Complete conformal metrics of constant positive scalar curvature with prescribed isolated singularities
AU - Hongyu Chen
AU - Li Ma
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31058
UR - https://arxiv.org/abs/2610.31058
AB - Let (M^n,g) be a smooth closed connected Riemannian manifold of dimension n≥3 with positive Yamabe constant. For any prescribed finite set Λ⊂ M containing at least two points, we construct a complete metric on M∖Λ, conformal to g, with scalar curvature n(n−1). At each singularity, the local geometric term in the reduced action comes from the first nonzero homogeneous metric term of degree at most (n−2)/2 in finite conformal normal coordinates, or from the mass of the Green function metric when all such terms vanish. We completely solve Schoen's conjecture about singular Yamabe problem.
ER -