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Lower bounds for the first eigenvalue of the $p$-Laplacian on quaternionic Kähler manifolds

Kui Wang, Shaoheng Zhang

Abstract

We study the first nonzero eigenvalues for the $p$-Laplacian on quaternionic Kähler manifolds. Our first result is a lower bound for the first nonzero closed (Neumann) eigenvalue of the $p$-Laplacian on compact quaternionic Kähler manifolds. Our second result is a lower bound for the first Dirichlet eigenvalue of the $p$-Laplacian on compact quaternionic Kähler manifolds with smooth boundary. Our results generalize corresponding results for the Laplacian eigenvalues on quaternionic Kähler manifolds proved in [22].

Lower bounds for the first eigenvalue of the $p$-Laplacian on quaternionic Kähler manifolds

Abstract

We study the first nonzero eigenvalues for the -Laplacian on quaternionic Kähler manifolds. Our first result is a lower bound for the first nonzero closed (Neumann) eigenvalue of the -Laplacian on compact quaternionic Kähler manifolds. Our second result is a lower bound for the first Dirichlet eigenvalue of the -Laplacian on compact quaternionic Kähler manifolds with smooth boundary. Our results generalize corresponding results for the Laplacian eigenvalues on quaternionic Kähler manifolds proved in [22].
Paper Structure (5 sections, 12 theorems, 97 equations)

This paper contains 5 sections, 12 theorems, 97 equations.

Table of Contents

  1. Introduction and Main Results
  2. Quaternionic Kähler manifolds
  3. Modulus of Continuity Estimates
  4. The First Dirichlet Eigenvalue
  5. Elliptic Proof of Theorem \ref{['thm 1.1']}

Key Result

Theorem 1.1

Let $(M^m, g)$ be a compact Riemannian manifold (possibly with a smooth convex boundary) with diameter $D$ and $\operatorname{Ric} \geq (m-1)\kappa$ for $\kappa\in \mathbb{R}$. Let $\mu_{1}$ be the first nonzero eigenvalue of the Laplacian on $M$ (with Neumann boundary condition if $\partial M \neq where $\bar{\mu}_1(m,\kappa,D)$ is the first nonzero Neumann eigenvalue of the one-dimensional eige

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2: LW23
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6: LW23
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • ...and 11 more