Density and spectrum of minimal submanifolds in space forms

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 $Θ(r)$ of $M^m$ and the spectrum of the 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 $\mathbb{N}^m_k$. 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.

Theorems & Definitions (33)

• Lemma 1
• Remark 1
• Corollary 1
• Definition 1
• Theorem 1
• Corollary 2
• Remark 2
• Theorem 2
• Remark 3
• Proposition 1