Density and spectrum of minimal submanifolds in space forms

Let $M^m$ be a minimal properly immersed submanifold in an ambient space close, in a suitable sense, to the space form ${\mathbb{N}}_k^n$ of curvature $-k\le 0$. In this paper, we are interested in the relation between the density function $\mathrm{\Theta }(r)$ of $M^m$ and the spectrum of the Laplace-Beltrami operator. In particular, we prove that if $\mathrm{\Theta }(r)$ 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 ${\mathbb{N}}_k^m$. Notably, the result applies to Anderson’s (smooth) solutions of Plateau’s problem at infinity on the hyperbolic space ${\mathbb{H}}^n$, 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 ${\mathbb{H}}^n$ with finite total curvature have finite density.