Ends, Fundamental Tones, and Capacities of Minimal Submanifolds Via Extrinsic Comparison Theory
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.
KeywordsFirst Dirichlet eigenvalue Capacity Effective resistance Minimal submanifolds Fundamental tone Minimal submanifolds
Mathematics Subject Classification (2010)53A 53C
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