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arXiv · 2610.31058

Complete conformal metrics of constant positive scalar curvature with prescribed isolated singularities

Hongyu Chen·Li Ma
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Abstract

Let $(M^n,g)$ be a smooth closed connected Riemannian manifold of dimension $n\ge3$ with positive Yamabe constant. For any prescribed finite set $Λ\subset M$ containing at least two points, we construct a complete metric on $M\setminusΛ$, 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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Hongyu Chen, Li Ma. 2026-10-22. Complete conformal metrics of constant positive scalar curvature with prescribed isolated singularities. https://arxiv.org/abs/2610.31058

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