Spectrum of the Laplacian on Quaternionic Kahler Manifolds

Abstract

Let $M^{4n}$ be a complete quaternionic Kähler manifold with scalar curvature bounded below by $-16n(n+2)$. We get a sharp estimate for the first eigenvalue ${\lambda }_1(M)$ of the Laplacian which is ${\lambda }_1(M)\le (2n+1{)}^2$. If the equality holds, then either $M$ has only one end, or $M$ is diffeomorphic to $\mathbb{R}\times N$ with N given by a compact manifold. Moreover, if $M$ is of bounded curvature, $M$ is covered by the quaterionic hyperbolic space ${\mathbb{QH}}^n$ and $N$ is a compact quotient of the generalized Heisenberg group. When ${\lambda }_1(M)\ge \frac{8(n+2)}{3}$, we also prove that $M$ must have only one end with infinite volume.

arXiv comment46 pages

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