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On the essential spectra of submanifolds in the hyperbolic space

Gerasim Kokarev

TL;DR

The paper analyzes how the asymptotic geometry of submanifolds in hyperbolic space $\mathbf{H}^{n+1}$ controls regularity at the ideal boundary and the spectrum of the Laplacian. It introduces regularity at infinity and asymptotically minimal submanifolds, proving that minimal submanifolds extending to a $C^1$-smooth boundary meet the boundary orthogonally and that the essential spectrum of such submanifolds contains $[\frac{(m-1)^2}{4},+\infty)$. The author develops tangent-cone analysis at ideal-boundary points and establishes lower bounds for the bottom of the essential spectrum via a Cheeger-constant estimate for annuli, which extend to Cartan-Hadamard ambient spaces with curvature not exceeding $-1$. The results provide a stability statement: the essential spectrum is preserved under asymptotically minimal perturbations, aligning with the extremal behavior when discrete spectrum is absent. Together, these findings link boundary regularity, mean curvature decay, and spectral properties in a coherent framework for hyperbolic submanifolds.

Abstract

We study relationships between asymptotic geometry of submanifolds in the hyperbolic space and their regularity properties near the ideal boundary, revisiting some of the related results in the literature. In particular, we discuss hypotheses when minimal submanifolds meet the ideal boundary orthogonally, and compute the essential spectrum of the Laplace operator on submanifolds that are asymptotically close to minimal submanifolds.

On the essential spectra of submanifolds in the hyperbolic space

TL;DR

The paper analyzes how the asymptotic geometry of submanifolds in hyperbolic space controls regularity at the ideal boundary and the spectrum of the Laplacian. It introduces regularity at infinity and asymptotically minimal submanifolds, proving that minimal submanifolds extending to a -smooth boundary meet the boundary orthogonally and that the essential spectrum of such submanifolds contains . The author develops tangent-cone analysis at ideal-boundary points and establishes lower bounds for the bottom of the essential spectrum via a Cheeger-constant estimate for annuli, which extend to Cartan-Hadamard ambient spaces with curvature not exceeding . The results provide a stability statement: the essential spectrum is preserved under asymptotically minimal perturbations, aligning with the extremal behavior when discrete spectrum is absent. Together, these findings link boundary regularity, mean curvature decay, and spectral properties in a coherent framework for hyperbolic submanifolds.

Abstract

We study relationships between asymptotic geometry of submanifolds in the hyperbolic space and their regularity properties near the ideal boundary, revisiting some of the related results in the literature. In particular, we discuss hypotheses when minimal submanifolds meet the ideal boundary orthogonally, and compute the essential spectrum of the Laplace operator on submanifolds that are asymptotically close to minimal submanifolds.

Paper Structure

This paper contains 17 sections, 13 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Main results and discussion
  3. Organisation of the paper
  4. Tangent cones at points on ideal boundary
  5. Preliminary material
  6. Proof of Theorem \ref{['t1']}
  7. Other results and extensions
  8. An estimate for the bottom of essential spectrum
  9. An inequality for the Cheeger constant
  10. Proof of Theorem \ref{['tc']}
  11. Examples of asymptotically minimal submanifolds
  12. Relationships with $C^2$-regularity
  13. Discussion and other consequences
  14. Proof of Theorem \ref{['t2']}
  15. Outline of the strategy
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\Sigma^m\subset\mathbf{H}^{n+1}$ be a minimal submanifold that extends to a $C^1$-smooth submanifold with boundary $\Gamma^{m-1}\subset S^n_\infty(\mathbf H^{n+1})$ in the closure $\bar{\mathbf H}^{n+1}$. Then $\Sigma^m$ meets the sphere at infinity $S^n_\infty(\mathbf H^{n+1})$ orthogonally.

Theorems & Definitions (28)

  • Definition 1.1
  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.2
  • Proposition 2.1
  • proof : Outline of the proof of Proposition \ref{['dist:prop']}
  • Claim 2.2
  • proof
  • Claim 2.3
  • proof
  • ...and 18 more