Abstract
Let \((X_1, \bar{g}_1)\) and \((X_2, \bar{g}_2)\) be two compact Riemannian manifolds with boundary \((M_1,g_1)\) and \((M_2,g_2)\), respectively. The Escobar problem consists in prescribing a conformal metric on a compact manifold with boundary with zero scalar curvature in the interior and constant mean curvature of the boundary. The present work is the construction of a connected sum \(X=X_1 \#X_2\) by excising half ball near points on the boundary. The resulting metric on X has zero scalar curvature and a CMC boundary. We fully exploit the non-local aspect of the problem and use new tools developed in recent years to handle such kinds of issues. Our problem is of course a very well-known problem in geometric analysis and that is why we consider it but the results in the present paper can be extended to other more analytical problems involving connected sums of constant fractional curvatures.
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Acknowledgements
Part of the work was done when W. Ao was visiting the math department of Universidad Autónoma de Madrid in June 2018, she thanks the hospitality of the department. W. Ao was supported by NSFC (No. 11801421 and No. 11631011). M.d.M. González is supported by the Spanish Government Grants MTM2014-52402-C3-1-P and MTM2017-85757-P. Y. Sire would like to thank E. Delay and F. Pacard for useful discussions.
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Ao, W., González, M.d.M. & Sire, Y. Boundary Connected Sum of Escobar Manifolds. J Geom Anal 30, 4092–4109 (2020). https://doi.org/10.1007/s12220-019-00231-1
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DOI: https://doi.org/10.1007/s12220-019-00231-1