Eigenvalue Estimates for the Bochner Laplacian and Harmonic Forms on Complete Manifolds

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.