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

We give a fairly general answer for a question of S. T. Yau in his millennium lectures \cite[p.240]{yau} about what upper bounds for the first eigenvalue of properly immersed minimal surfaces of $\mathbb{R}^{3}$ in terms of its volume growth. In fact, 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. This result can be viewed as an extrinsic version of Brook's essential spectrum estimate \cite[Thm.1]{brooks}.