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Sharp estimates for the Robin Laplacian under a perimeter constraint in hyperbolic space

Daguang Chen, Shan Li

TL;DR

The paper investigates Robin Laplacian eigenvalues on horospherically convex domains in hyperbolic space under a perimeter constraint. It develops a parallel-coordinate framework together with hyperbolic Steiner-type and Alexandrov-Fenchel tools to compare a domain with a perimeter-matched geodesic ball. For negative boundary parameter β, it establishes a lower-bound deficit that implies the geodesic ball maximizes the first eigenvalue, with equality only for balls; for positive β it provides complementary upper-bound deficits, connecting to Faber-Krahn-type phenomena. The results extend known spectral-isoperimetric inequalities to hyperbolic geometry and furnish explicit deficit controls in terms of isoperimetric deficit and extremal ball eigenfunctions.

Abstract

In this paper, we establish a lower bound, in terms of the isoperimetric deficit, for the first eigenvalue of the Robin Laplacian with negative boundary parameter on horospherically convex bounded domains in hyperbolic space, which implies that the geodesic ball maximizes this eigenvalue among all such domains. Furthermore, we derive upper bounds for the first eigenvalue of the Robin Laplacian with positive boundary parameter on horospherically convex bounded domains in hyperbolic space.

Sharp estimates for the Robin Laplacian under a perimeter constraint in hyperbolic space

TL;DR

The paper investigates Robin Laplacian eigenvalues on horospherically convex domains in hyperbolic space under a perimeter constraint. It develops a parallel-coordinate framework together with hyperbolic Steiner-type and Alexandrov-Fenchel tools to compare a domain with a perimeter-matched geodesic ball. For negative boundary parameter β, it establishes a lower-bound deficit that implies the geodesic ball maximizes the first eigenvalue, with equality only for balls; for positive β it provides complementary upper-bound deficits, connecting to Faber-Krahn-type phenomena. The results extend known spectral-isoperimetric inequalities to hyperbolic geometry and furnish explicit deficit controls in terms of isoperimetric deficit and extremal ball eigenfunctions.

Abstract

In this paper, we establish a lower bound, in terms of the isoperimetric deficit, for the first eigenvalue of the Robin Laplacian with negative boundary parameter on horospherically convex bounded domains in hyperbolic space, which implies that the geodesic ball maximizes this eigenvalue among all such domains. Furthermore, we derive upper bounds for the first eigenvalue of the Robin Laplacian with positive boundary parameter on horospherically convex bounded domains in hyperbolic space.
Paper Structure (5 sections, 6 theorems, 51 equations)

This paper contains 5 sections, 6 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quermassintegrals
  4. Parallel hypersurfaces and Alexandrov-Fenchel inequality
  5. Proofs of Theorem \ref{['thm1']} and \ref{['thm4']}

Key Result

Theorem 1.1

Let $\Omega$ be a horospherically convex bounded domain with smooth boundary in hyperbolic space $\mathbb{H}^n$ and $P(\Omega)$ be the perimeter of $\Omega$. Let $\Omega^\star$ be the geodesic ball with the same perimeter as $\Omega$, that is, $P\left(\Omega\right)=P\left(\Omega^\star\right)$. For $ where $v_m=\min\limits_{x\in\Omega^\star}v(x)$ and $\left|\Omega\right|$ is the volume of $\Omega$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Corollary 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Theorem 2.1
  • Lemma 2.1
  • ...and 3 more