Every closed four-manifold with at least two Z2-conical points admits a Yamabe metric, a conformal metric of constant scalar curvature, via a new min-max argument.
Conformal Green functions and Yamabe metrics of Sobolev regularity
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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 $n \geq 3$ and for $W^{2,q}$ Riemannian metrics with $q>\tfrac{n}{2}$ and should be of independent interest.
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math.DG 1years
2025 1verdicts
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Min-max theory and Yamabe metrics on conical four-manifolds
Every closed four-manifold with at least two Z2-conical points admits a Yamabe metric, a conformal metric of constant scalar curvature, via a new min-max argument.