On the fundamental tone of minimal submanifolds with controlled extrinsic curvature

TL;DR

This paper investigates bounds on the fundamental tone of minimal submanifolds in Euclidean and hyperbolic spaces, establishing conditions on extrinsic curvature and volume growth that determine spectral properties.

Contribution

It provides new sufficient conditions on extrinsic curvature and volume growth to determine the fundamental tone bounds of minimal submanifolds in certain ambient spaces.

Findings

Established bounds for the fundamental tone under curvature restrictions

Identified conditions on the second fundamental form for spectral bounds

Derived intrinsic volume growth conditions for spectral estimates

Abstract

The aim of this paper is to obtain the fundamental tone for minimal submanifolds of the Euclidean or hyperbolic space under certain restrictions on the extrinsic curvature. We show some sufficient conditions on the norm of the second fundamental form that allow us to obtain the same upper and lower bound for the fundamental tone of minimal submanifolds in a Cartan-Hadamard ambient manifold. As an intrinsic result, we obtain a sufficient condition on the volume growth of a Cartan-Hadamard manifold to achieve the lowest bound for the fundamental tone given by McKean.