Eigenvalue estimates for the Bochner Laplacian and harmonic forms on complete manifolds Nelia Charalambous 58J6035P15Bochner LaplacianHodge Laplacianharmonic formsSobolev inequality 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. Indiana University Mathematics Journal 2010 text pdf 10.1512/iumj.2010.59.3770 10.1512/iumj.2010.59.3770 en Indiana Univ. Math. J. 59 (2010) 183 - 206 state-of-the-art mathematics http://iumj.org/access/