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Remarks on the spectra of minimal hypersurfaces in the hyperbolic space

Gerasim Kokarev

TL;DR

The paper establishes that singular area-minimising currents in hyperbolic space with prescribed asymptotic boundary have Laplace spectra starting at the same critical value as their hyperbolic model. By developing a sharp isoperimetric comparison and leveraging boundary regularity results, the authors prove that for any area-minimising current in the natural class, the fundamental tone is at most $\frac{(m-1)^2}{4}$, with equality attained by a complete current $\Sigma^m$ in the same class. They then show that the spectrum of such an area-minimising current is the full interval $\big[(m-1)^2/4,+\infty\big)$, by constructing Weyl-type sequences and establishing essential spectrum bounds. The results extend to higher codimension and rely on a combination of McKean-type bounds, Anderson–Lin regularity, and careful analysis near the ideal boundary, without requiring strong curvature or volume growth assumptions. This sharp spectral characterization links asymptotic Plateau problems to spectral geometry in hyperbolic spaces and highlights the role of boundary regularity in determining self-adjoint extensions and spectral properties.

Abstract

We compute the Laplacian spectra of singular area-minimising hypersurfaces in the hyperbolic space with prescribed asymptotic data. We also obtain similar results in higher codimension, and explore related extremal properties of the bottom of the spectrum.

Remarks on the spectra of minimal hypersurfaces in the hyperbolic space

TL;DR

The paper establishes that singular area-minimising currents in hyperbolic space with prescribed asymptotic boundary have Laplace spectra starting at the same critical value as their hyperbolic model. By developing a sharp isoperimetric comparison and leveraging boundary regularity results, the authors prove that for any area-minimising current in the natural class, the fundamental tone is at most , with equality attained by a complete current in the same class. They then show that the spectrum of such an area-minimising current is the full interval , by constructing Weyl-type sequences and establishing essential spectrum bounds. The results extend to higher codimension and rely on a combination of McKean-type bounds, Anderson–Lin regularity, and careful analysis near the ideal boundary, without requiring strong curvature or volume growth assumptions. This sharp spectral characterization links asymptotic Plateau problems to spectral geometry in hyperbolic spaces and highlights the role of boundary regularity in determining self-adjoint extensions and spectral properties.

Abstract

We compute the Laplacian spectra of singular area-minimising hypersurfaces in the hyperbolic space with prescribed asymptotic data. We also obtain similar results in higher codimension, and explore related extremal properties of the bottom of the spectrum.
Paper Structure (14 sections, 11 theorems, 106 equations)

This paper contains 14 sections, 11 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Content and organisation of the paper
  4. Preliminary material
  5. Notation and first results
  6. Laplacian and its spectrum
  7. Comparison of the isoperimetric constants
  8. Main statement
  9. Proof of Theorem \ref{['tc']}
  10. Proofs
  11. Proof of Theorem \ref{['t2']}: the inequality
  12. Proof of Theorem \ref{['t2']}: the spectrum of the area-minimising current
  13. Proof of Theorem \ref{['t1']}
  14. Final remarks

Key Result

Theorem 1.1

Let $\Sigma^m\subset\mathbf H^{n+1}$ be a proper $C^2$-smooth minimal submanifold in the hyperbolic space $\mathbf H^{n+1}$ that extends to a $C^2$-smooth manifold with boundary in the closure ${\mathbf {\bar{H}}^{n+1}}$, obtained by adding the sphere at infinity $\mathbf H^{n+1}\cup S^n_\infty(\mat

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.1
  • ...and 13 more