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A Prescribed Scalar and Boundary Mean Curvature Problem and the Yamabe Classification on Asymptotically Euclidean Manifolds with Inner Boundary

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Abstract

We consider the problem of finding a metric in a given conformal class with prescribed non-positive scalar curvature and non-positive boundary mean curvature on an asymptotically Euclidean manifold with inner boundary. We obtain a necessary and sufficient condition in terms of a conformal invariant of the zero sets of the target curvatures for the existence of solutions to the problem and use this result to establish the Yamabe classification of metrics in those manifolds with respect to the solvability of the prescribed curvature problem.

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References

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Acknowledgements

We would like to thank David Maxwell for the lead on the proof of Lemma 5.6. This work is supported by an NSERC Discovery Grant. The second author was also supported by the fellowship grant P2021-4201 of the National University of Mongolia.

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Authors and Affiliations

  1. McGill University, Montreal, Canada

    Vladmir Sicca & Gantumur Tsogtgerel

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  1. Vladmir Sicca
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  2. Gantumur Tsogtgerel
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Correspondence to Vladmir Sicca.

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Sicca, V., Tsogtgerel, G. A Prescribed Scalar and Boundary Mean Curvature Problem and the Yamabe Classification on Asymptotically Euclidean Manifolds with Inner Boundary. J Geom Anal 33, 342 (2023). https://doi.org/10.1007/s12220-023-01346-2

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  • DOI: https://doi.org/10.1007/s12220-023-01346-2

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