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 R n in terms of their volume growth. Our result can be viewed as an extrinsic version of Brooks's essential spectrum estimate (Brooks, Math Z 178(4): 501-508, 1981, Thm. 1) and it gives a fairly general answer to a question of Yau (Asian J Math 4(1): 2000) about upper bounds for the first eigenvalue (bottom of the spectrum) of immersed minimal surfaces of R 3 .