## Journal of Differential Geometry

- J. Differential Geom.
- Volume 99, Number 2 (2015), 215-253.

### On the spectrum of bounded immersions

G. Pacelli Bessa, Luquésio P. Jorge, and Luciano Mari

#### Abstract

In this article, we investigate some of the relations between the spectrum of a non-compact, extrinsically bounded submanifold $\varphi : M^m \to N^n$ and the Hausdorff dimension of its limit set $\lim \varphi$. In particular, we prove that if $\varphi : M^2 \to \mathbb{R}^3$ is a (complete) minimal surface immersed into an open, bounded, strictly convex subset $\Omega$ with $C^3$-boundary, then $M$ has discrete spectrum, provided that $\mathcal{H}_{\Psi} (\lim \varphi \cap \Omega) = 0$, where $\mathcal{H}_{\Psi}$ is the Hausdorff measure of order $\Psi(t) = t^2 \lvert \log t \rvert$. Our main theorem, Thm. 2.4, applies to a number of examples recently constructed, by many authors, in the light of Nadirashvili’s discovery of complete bounded minimal disks in $\mathbb{R}^3$, as well as to solutions of Plateau’s problems for non-rectifiable Jordan curves, giving a fairly complete answer to a question posed by S.T. Yau in his Millenium Lectures.

On the other hand, we present a simple criterion, called *the ball property,* whose fulfilment guarantees the existence of elements in the essential spectrum. As an application, we show that some of the examples
of Jorge-Xavier and Rosenberg-Toubiana of complete minimal surfaces between two planes have essential spectrum $\sigma_{\mathrm{ess}}(-\Delta) = [0, \infty)$.

#### Article information

**Source**

J. Differential Geom., Volume 99, Number 2 (2015), 215-253.

**Dates**

First available in Project Euclid: 16 January 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1421415562

**Digital Object Identifier**

doi:10.4310/jdg/1421415562

**Mathematical Reviews number (MathSciNet)**

MR3302039

**Zentralblatt MATH identifier**

1327.53076

#### Citation

Bessa, G. Pacelli; Jorge, Luquésio P.; Mari, Luciano. On the spectrum of bounded immersions. J. Differential Geom. 99 (2015), no. 2, 215--253. doi:10.4310/jdg/1421415562. https://projecteuclid.org/euclid.jdg/1421415562