Eigenvalue estimates for 3-Sasaki structures

TL;DR

This paper establishes new lower bounds for the first non-zero eigenvalue of the scalar sub-Laplacian on 3-Sasaki manifolds, generalizing classical estimates and providing explicit eigenfunctions in certain cases.

Contribution

It introduces improved eigenvalue estimates for 3-Sasaki structures, describes the limiting eigenspaces, and computes second eigenvalues in dimension 7, extending known geometric analysis results.

Findings

New lower bounds for the first eigenvalue of the sub-Laplacian.

Explicit eigenfunctions for the second eigenvalue in dimension 7.

Lower bounds for the Riemannian Laplacian spectrum in the canonical variation.

Abstract

We obtain new lower bounds for the first non-zero eigenvalue of the scalar sub-Laplacian for 3-Sasaki metrics, improving Lichnerowicz-Obata type estimates by Ivanov et al. The limiting eigenspace is fully decribed in terms of the automorphism algebra. Our results can be thought of as an analogue of the Lichnerowicz-Matsushima estimate for K\"ahler-Einstein metrics. In dimension 7, if the automorphism algebra is non-vanishing, we also compute the second eigenvalue for the sub-Laplacian and construct explicit eigenfunctions. In addition, for all metrics in the canonical variation of the 3-Sasaki metric we give a lower bound for the spectrum of the Riemannian Laplace operator, depending only on scalar curvature and dimension. We also strengthen a result pertaining to the growth rate of harmonic functions, due to Conlon, Hein and Sun, in the case of hyperk\"ahler cones. In this setup we also describe the space of holomorphic functions.