Ends, fundamental tones and capacity of minimal submanifolds via extrinsic comparison theory

Vicent Gimeno, Steen Markvorsen

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Abstract

We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entities in a rotationally symmetric model manifold. Using the asymptotic behavior of the volumes and capacities we then obtain upper bounds for the number of ends as well as estimates for the fundamental tone of the submanifolds in question.
Original languageEnglish
JournalPotential Analysis
Volume42
Issue number4
Pages (from-to)749-774
ISSN0926-2601
DOIs
Publication statusPublished - 2015

Keywords

  • First Dirichlet eigenvalue
  • Capacity
  • Effective resistance
  • Minimal submanifolds
  • Fundamental tone

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