Density and spectrum of minimal submanifolds in space forms

Let M be a minimal, properly immersed submanifold in a space form of non-positive curvature. In this paper, we are interested in the relation between the density function of M and the spectrum of the Laplace-Beltrami operator. In particular, we prove that if the density grows subexponentially at infinity (when the ambient manifold is hyperbolic space) or sub-polynomially (when the ambient manifold is Euclidean space) along a sequence, then the spectrum of M is the same as that of a totally geodesic slice of the same dimension as M. Notably, the result applies to Anderson's (smooth) solutions of Plateau's problem at infinity on the hyperbolic space, independently of their boundary regularity. We also give a simple condition on the second fundamental form that ensures M to have finite density. In particular, we show that minimal submanifolds of the hyperbolic space that have finite total curvature have also finite density.