Eigenvalue estimates for the Bochner Laplacian and harmonic forms on complete manifolds

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We study the set of eigenvalues of the Bochner Laplacian on a geodesic ball of an open manifold M, and find lower estimates for these eigenvalues when M satisfies a Sobolev inequality. We show that we can use these estimates to demonstrate that the set of harmonic forms of polynomial growth over M is finite dimensional, under sufficient curvature conditions. We also study in greater detail the dimension of the space of bounded harmonic forms on coverings of compact manifolds.