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

TL;DR

This paper investigates the geometric and spectral properties of minimal submanifolds immersed in ambient spaces with controlled curvature, providing comparison results and bounds on ends and fundamental tones.

Contribution

It introduces extrinsic comparison techniques to estimate volumes, capacities, and spectral bounds of minimal submanifolds in curved ambient spaces.

Findings

Upper bounds for the number of ends of minimal submanifolds

Estimates for the fundamental tone based on curvature comparisons

Comparison of volumes and capacities with model manifolds

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.