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On Free-Boundary Minimal Surfaces in the Riemannian Schwarzschild Manifold

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Abstract

Is it possible to obtain unbounded minimal surfaces in certain asymptotically flat three-manifolds as a limit of solutions to a natural mountain pass problem with diverging boundaries? In this work, we give evidence that this might be true by analyzing related aspects in the case of the exact Riemannian Schwarzschild manifold. More precisely, we observe that the simplest minimal surface in this space has Morse index one. We prove also a relationship between the length of the boundary and the density at infinity of general minimal surfaces satisfying a free-boundary condition along the horizon.

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Acknowledgements

Part of the research was carried out and finalized during the author’s postdoctoral period at Princeton University.

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  1. Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, Av. Humberto Monte s/n, 60455-760, Fortaleza, CE, Brazil

    Rafael Montezuma

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  1. Rafael Montezuma
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Correspondence to Rafael Montezuma.

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Montezuma, R. On Free-Boundary Minimal Surfaces in the Riemannian Schwarzschild Manifold. Bull Braz Math Soc, New Series 52, 1055–1071 (2021). https://doi.org/10.1007/s00574-021-00245-w

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  • DOI: https://doi.org/10.1007/s00574-021-00245-w

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