Eigenvalue Estimates on quaternion-K\"ahler Manifolds

TL;DR

This paper establishes lower bounds for the first eigenvalues of the Laplacian on compact quaternion-K"ahler manifolds, linking spectral properties to geometric data using heat equation techniques.

Contribution

It introduces new eigenvalue estimates for quaternion-K"ahler manifolds based on heat equation analysis and geometric comparison theorems.

Findings

Lower bounds for the first nonzero eigenvalues in terms of dimension, diameter, and scalar curvature.

Extension of eigenvalue estimates to Dirichlet boundary conditions.

Application of heat equation modulus of continuity to geometric spectral bounds.

Abstract

We prove lower bound for the first closed or Neumann nonzero eigenvalue of the Laplacian on a compact quaternion-K\"ahler manifold in terms of dimension, diameter, and scalar curvature lower bound. It is derived as large time implication of the modulus of continuity estimates for solutions of the heat equation. We also establish lower bound for the first Dirichlet eigenvalue in terms of geometric data, via a Laplace comparison theorem for the distance to the boundary function.