Abstract
We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of ℝ3 is discrete. This gives a positive answer to a question of Yau (Asian J. Math. 4:235–278, 2000).
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The first and third author were partially supported by CNPq-CAPES, Brazil.
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Bessa, G.P., Jorge, L.P. & Montenegro, J.F. The Spectrum of the Martin-Morales-Nadirashvili Minimal Surfaces Is Discrete. J Geom Anal 20, 63–71 (2010). https://doi.org/10.1007/s12220-009-9101-z
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DOI: https://doi.org/10.1007/s12220-009-9101-z