Density and spectrum of minimal submanifolds in space forms

Let be a minimal, proper immersion in an ambient space suitably close to a space form of curvature . In this paper, we are interested in the relation between the density function of M and the spectrum of its Laplace–Beltrami operator. In particular, we prove that if has subexponential growth (when ) or sub-polynomial growth () along a sequence, then the spectrum of is the same as that of the space form . 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.