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

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}.

After we posted the former version of the article in Arxiv, Ilias-Nelli-Soret pointed out that their had an extrinsic Brook's upper estimate for the essential spectrum of minimal submanifolds. In this replacements we give the due credits for their work. We are grateful they pointed it out to us