Spectrum of the Laplacian on Quaternionic Kahler Manifolds

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 $λ_1(M)$ of the Laplacian which is $λ_1(M)\le (2n+1)^2$. 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 $λ_1(M)\ge \frac{8(n+2)}3$, we also prove that $M$ must have only one end with infinite volume.

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