Eigenvalue estimates for 3-Sasaki structures

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 et al. in [16, 17]. 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 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&Sun [12, 15], in the case of hyperkähler cones. In this setup we also describe the space of holomorphic functions. 2020 Mathematics Subject Classification: Primary 53C25, 53C26, 58C40, 35H10.

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