Density and spectrum of minimal submanifolds in space forms

TL;DR

This paper investigates the relationship between the density function and the spectrum of minimal submanifolds in space forms, showing conditions under which their spectral properties match those of the ambient space.

Contribution

It establishes new links between density growth conditions and spectral equivalence for minimal submanifolds in space forms, including hyperbolic space.

Findings

Spectral equivalence under subexponential or sub-polynomial density growth.

Application to solutions of Plateau's problem at infinity in hyperbolic space.

Finite density for minimal submanifolds with finite total curvature.

Abstract

Let $M^m$ be a minimal properly immersed submanifold in an ambient space close, in a suitable sense, to the space form $\mathbb{N}^n_k$ of curvature $-k\le 0$. In this paper, we are interested in the relation between the density function $\Theta(r)$ of $M^m$ and the spectrum of the Laplace-Beltrami operator. In particular, we prove that if $\Theta(r)$ 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 $\mathbb{N}^m_k$. 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.