Abstract. Let ϕ : M m → N n be a minimal, proper immersion in an ambient space suitably close to a space form N n k of curvature −k ≤ 0. In this paper, we are interested in the relation between the density function Θ(r) of M and the spectrum of its Laplace-Beltrami operator. In particular, we prove that if Θ(r) has subexponential growth (when k < 0) or sub-polynomial growth (k = 0) along a sequence, then the spectrum of M m is the same as that of the space form N m k . 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 with finite total curvature in the hyperbolic space also have finite density. density and spectrum and Laplace-Beltrami and minimal submanifolds and monotonicity