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Renormalized Area for Minimal Hypersurfaces of 5D Poincaré–Einstein Spaces

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Abstract

In this paper we derive a Gauss–Bonnet formula for the renormalized area of Graham–Witten minimal hypersurfaces of 5-dimensional Poincaré–Einstein spaces. The formula we derive expresses the renormalized area in terms of integrals of local geometric quantities. We also prove a result which gives a characterization of minimal hypersurfaces with \(L^2\) second fundamental form in terms of conformal geometry at infinity.

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Acknowledgements

The majority of this work was carried out during the author’s Ph.D. studies at the University of Notre Dame. The author would like to thank his adviser Professor Matthew Gursky for introducing him to this area of geometry and to this problem in particular, as well as for many helpful conversations. The author would also like to thank his postdoctoral supervisor Professor Stephen McKeown for many helpful comments and for help surveying the literature. This work was partially supported by the National University of Ireland Éamon de Valera Travelling Studentship in Mathematics.

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  1. Mathematics and Statistics, Texas Tech University, Lubbock, TX, 79409, USA

    Aaron J. Tyrrell

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  1. Aaron J. Tyrrell
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Correspondence to Aaron J. Tyrrell.

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Tyrrell, A.J. Renormalized Area for Minimal Hypersurfaces of 5D Poincaré–Einstein Spaces. J Geom Anal 33, 310 (2023). https://doi.org/10.1007/s12220-023-01358-y

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