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The Yamabe Problem for Distributional Curvature

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Abstract

In this paper, we consider Yamabe problem for a smooth manifold with a \(W^{1,p}\) metric. The curvature is considered as a distribution since our metric is not twice differentiable. We prove that for any metric g which is \(W^{1,p}\) on M such that the Yamabe constant of (M, g) is less than that of the standard sphere, there exists a \(W^{1,p}\) metric on M which is conformal to g and has constant distributional scalar curvature.

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Acknowledgements

The research was supported by the National Key R &D Program of China (No. 2022YFA1005500) and the National Key R &D Program of China (No. 2020YFA0713100). The author was supported in part by NSF in China (No. 12071425). The author would like to express his sincere gratitude to Wenshuai Jiang and Weimin Sheng for their guidance and encouragement, and to Xi Zhang and Ting Zhang for many usefulsuggestions. He also thanks Xucheng Yu for useful discussions.

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  1. School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Huaiyu Zhang

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Zhang, H. The Yamabe Problem for Distributional Curvature. J Geom Anal 33, 312 (2023). https://doi.org/10.1007/s12220-023-01366-y

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