The spectrum of the Laplacian and volume growth of proper minimal submanifolds

TL;DR

This paper establishes upper bounds for the essential spectrum of properly immersed minimal submanifolds in Euclidean space, linking spectral properties to volume growth, and improves existing estimates by previous researchers.

Contribution

It provides a refined extrinsic estimate for the bottom of the essential spectrum of minimal submanifolds based on their volume growth.

Findings

Upper bounds for the essential spectrum are derived in terms of volume growth.

The results improve upon previous estimates by Ilias-Nelli-Soret.

The work connects spectral theory with geometric volume growth properties.

Abstract

We give upper bounds for the bottom of the essential spectrum of properly immersed minimal submanifolds of $\mathbb{R}^{n}$ in terms of their volume growth. Our result improves the extrinsic version of Brook's essential spectrum estimate given by Ilias-Nelli-Soret in \cite[Cor.3]{ins}.