Journal of Geometric Analysis

, Volume 20, Issue 1, pp 63–71

The Spectrum of the Martin-Morales-Nadirashvili Minimal Surfaces Is Discrete

  • G. Pacelli Bessa
  • Luquesio P. Jorge
  • J. Fabio Montenegro
Article

DOI: 10.1007/s12220-009-9101-z

Cite this article as:
Bessa, G.P., Jorge, L.P. & Montenegro, J.F. J Geom Anal (2010) 20: 63. doi:10.1007/s12220-009-9101-z

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).

Keywords

Proper bounded minimal submanifoldsDiscrete spectrumEssential spectrum

Mathematics Subject Classification (2000)

53C4053C4258C40

Copyright information

© Mathematica Josephina, Inc. 2009

Authors and Affiliations

  • G. Pacelli Bessa
    • 1
  • Luquesio P. Jorge
    • 1
  • J. Fabio Montenegro
    • 1
  1. 1.Department of MathematicsUniversidade Federal do Ceara–UFCFortalezaBrazil