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On the spectrum of bounded immersions

Gregorio Pacelli Bessa, Luquesio P. Jorge, Luciano Mari

Abstract

In this paper, we investigate the relationship between the discreteness of the spectrum of a non-compact, extrinsically bounded submanifold $\varphi \colon M^m \ra N^n$ and the Hausdorff dimension of its limit set $\lim\varphi$. In particular, we prove that if $\varphi \colon \!M^2 \ra D \subseteq \R^3$ is a minimal immersion into an open, bounded, strictly convex subset $D$ with $C^2$-boundary, then $M$ has discrete spectrum provided that $\haus_Ψ(\lim\varphi \cap D)=0$, where $\haus_Ψ$ is the generalized Hausdorff measure of order $Ψ(t) = t^2|\log t|$. Our theorem applies to a number of examples recently constructed by various authors in the light of N. Nadirashvili's discovery of complete, bounded minimal disks in $\R^3$, as well as to solutions of Plateau's problems, giving a fairly complete answer to a question posed by S.T. Yau in his Millenium Lectures. Suitable counter-examples show the sharpness of our results: in particular, we develop a simple criterion for the existence of essential spectrum which is suited for the techniques developed after Jorge-Xavier and Nadirashvili's examples.

On the spectrum of bounded immersions

Abstract

In this paper, we investigate the relationship between the discreteness of the spectrum of a non-compact, extrinsically bounded submanifold and the Hausdorff dimension of its limit set . In particular, we prove that if is a minimal immersion into an open, bounded, strictly convex subset with -boundary, then has discrete spectrum provided that , where is the generalized Hausdorff measure of order . Our theorem applies to a number of examples recently constructed by various authors in the light of N. Nadirashvili's discovery of complete, bounded minimal disks in , as well as to solutions of Plateau's problems, giving a fairly complete answer to a question posed by S.T. Yau in his Millenium Lectures. Suitable counter-examples show the sharpness of our results: in particular, we develop a simple criterion for the existence of essential spectrum which is suited for the techniques developed after Jorge-Xavier and Nadirashvili's examples.

Paper Structure

This paper contains 4 sections, 11 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Setting, notations and preliminaries
  3. The main theorem and its proof
  4. Remarks on $\sigma_\mathrm{ess}(-\Delta)$ and open questions

Key Result

Theorem 1

Let $\varphi \colon M^2 \to D \subset N$ be a minimal immersion into an open, bounded, $2$-convex subset of a Cartan-Hadamard manifold $N$. Set $\Psi(t) = t^2|\log t|$. If the $\Psi$-Hausdorff measure of $\lim\varphi \cap D$ satisfies $\mathcal{H}_\Psi(\lim\varphi \cap D) = 0$, then the spectrum of

Theorems & Definitions (34)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Corollary 1
  • Corollary 2
  • Remark 3
  • Example 1
  • Remark 4
  • Definition 1
  • Remark 5
  • ...and 24 more