Calculus of Variations and Geometric Measure Theory

B. P. Lima - J. F. Montenegro - L. Mari - F. B. Vieira

Density and spectrum of minimal submanifolds in space forms

created by mari1 on 04 Jan 2021



Inserted: 4 jan 2021
Last Updated: 4 jan 2021

Journal: Math. Ann.
Volume: 366
Number: 3-4
Pages: 1035-1066
Year: 2016
Doi: 10.1007/s00208-016-1360-y

ArXiv: 1407.5280 PDF


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.