Generalized Bochner theorems and the spectrum of complete manifolds (Q1123467)

The results of the present paper are centered around generalizations of the well-known Bochner theorem to manifolds where the Ricci curvature is bounded below by a negative constant C when \(C\geq -\lambda_ 0\), for \(\lambda_ 0\) the infimum of the spectrum of the Laplacian acting on \(L^ 2\) functions on M. The authors obtain vanishing theorems for square integrable harmonic one-forms and for cohomology with compact support. They also obtain vanishing results for harmonic p-forms given similar bounds on \(\lambda_ 0\). More refined results are obtained by taking a certain Brownian motion average of the Ricci curvature.