Conformal Green functions and Yamabe metrics of Sobolev regularity

2025/07/02 by Avalos, Rodrigo, Cogo, Albachiara, Abrego, Andoni Royo · 2 citations
#35J08 #35J20 #35J61 #53C18 #58J05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.01674

Abstract

We provide a full resolution of the Yamabe problem on closed 3-manifolds for Riemannian metrics of Sobolev class W^2,q with q > 3. This requires developing an elliptic theory for the conformal Laplacian for rough metrics and establishing existence, regularity and a delicate blow-up analysis for its Green function. Most of the analytical work is carried out in dimensions n ≥ 3 and for W^2,q Riemannian metrics with q>\tfracn2 and should be of independent interest.

Cited by

• Regularity of conformal structures on closed 3-manifolds
• Min-max theory and Yamabe metrics on conical four-manifolds

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