Eigenvalue estimates for 3-Sasaki structures

Abstract

Abstract We obtain new lower bounds for the first non-zero eigenvalue of the scalar sub-Laplacian for 3-Sasaki metrics, improving the Lichnerowicz–Obata-type estimates by Ivanov, Petkov and Vassilev (2013, 2014). The limiting eigenspace is fully described in terms of the automorphism algebra. Our results can be thought of as an analogue of the Lichnerowicz–Matsushima estimate for Kähler–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 (2013, 2017), in the case of hyperkähler cones. In this setup we also describe the space of holomorphic functions.