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On Eigenvalues of Logarithmic Potential Operator in the Hyperbolic Space

Jiya Rose Johnson, Sheela Verma

TL;DR

"${\mathcal{L}}_h$" extends the Euclidean logarithmic potential to hyperbolic geometry on the Poincaré disk, with a polarization scheme built from geodesic reflections. The authors prove a hyperbolic reverse Faber–Krahn inequality for the largest eigenvalue ${\tau}_h(\Omega)$ under polarization and provide an explicit eigenfunction representation; they also show ${\mathcal{L}}_h$ is a positive, compact, self-adjoint operator and that $0$ is not an eigenvalue. The work generalizes classical Euclidean results to hyperbolic space by leveraging Möbius invariance of distances and reflection symmetries. These results enrich spectral theory of hyperbolic integral operators and potential theory on Riemannian manifolds, with implications for eigenfunction structure and extremal domain analysis."

Abstract

Let $Ω$ be a bounded open set in the Poincaré hyperbolic disk, $\mathbb{D}$. In this article, we consider the hyperbolic logarithmic potential operator $\mathcal{L}_h : L^2(Ω) \to L^2(Ω)$, defined by \begin{equation*} \mathcal{L}_h u(z)=\frac{1}{2}\int_Ω\log\frac{1}{[z,w]}\,u(w)\, {\,\rm d}(w), \end{equation*} and the associated eigenvalue problem on $Ω$ \begin{equation} \mathcal{L}_h u=τu. \end{equation} We first extend the notion of polarization with respect to hyperplanes in the Poincaré disk and prove the associated properties. Then we establish a reverse Faber-Krahn inequality for the largest eigenvalue, $τ_{h}$ of $\mathcal{L}_h$, under polarization. Further, we provide a representation formula for the eigenfunctions of $\mathcal{L}_h$. In addition, we show that the operator $\mathcal{L}_h$ is a positive operator on $L^2(Ω)$.

On Eigenvalues of Logarithmic Potential Operator in the Hyperbolic Space

TL;DR

"" extends the Euclidean logarithmic potential to hyperbolic geometry on the Poincaré disk, with a polarization scheme built from geodesic reflections. The authors prove a hyperbolic reverse Faber–Krahn inequality for the largest eigenvalue under polarization and provide an explicit eigenfunction representation; they also show is a positive, compact, self-adjoint operator and that is not an eigenvalue. The work generalizes classical Euclidean results to hyperbolic space by leveraging Möbius invariance of distances and reflection symmetries. These results enrich spectral theory of hyperbolic integral operators and potential theory on Riemannian manifolds, with implications for eigenfunction structure and extremal domain analysis."

Abstract

Let be a bounded open set in the Poincaré hyperbolic disk, . In this article, we consider the hyperbolic logarithmic potential operator , defined by \begin{equation*} \mathcal{L}_h u(z)=\frac{1}{2}\int_Ω\log\frac{1}{[z,w]}\,u(w)\, {\,\rm d}(w), \end{equation*} and the associated eigenvalue problem on \begin{equation} \mathcal{L}_h u=τu. \end{equation} We first extend the notion of polarization with respect to hyperplanes in the Poincaré disk and prove the associated properties. Then we establish a reverse Faber-Krahn inequality for the largest eigenvalue, of , under polarization. Further, we provide a representation formula for the eigenfunctions of . In addition, we show that the operator is a positive operator on .
Paper Structure (3 sections, 17 theorems, 144 equations, 3 figures)

This paper contains 3 sections, 17 theorems, 144 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Polarization on poincaré disk model
  3. Proof of the main theorems

Key Result

Theorem 1.2

Let $\Omega$ be an open bounded subset of Poincaré hyperbolic disk $\mathbb{D}$. Let $\mathcal{G}$ be a geodesic in $\mathbb{D}$ and let $\mathcal{H}$ be an associated polarizer. Then Moreover, equality in pollog holds if and only if where $\triangle$ denotes the symmetric difference of sets and $|\cdot|$ denotes the Lebesgue area measure.

Figures (3)

  • Figure 1: Geodesic passing through $(a,0)$
  • Figure 2: $T(U\cap {\mathbb D})={\mathbb D}\cap \{x>0\}$
  • Figure 3: $T(\overline{U}^c\cap {\mathbb D})={\mathbb D}\cap \{x<0\}$

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • ...and 26 more