Elworthy, K. D.; Rosenberg, Steven Generalized Bochner theorems and the spectrum of complete manifolds. (English) Zbl 0677.58046 Acta Appl. Math. 12, No. 1, 1-33 (1988). 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. Reviewer: S.K.Chatterjea Cited in 2 ReviewsCited in 9 Documents MSC: 58J65 Diffusion processes and stochastic analysis on manifolds 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 58J35 Heat and other parabolic equation methods for PDEs on manifolds Keywords:hyperbolic manifolds; Bochner theorem; Laplacian PDFBibTeX XMLCite \textit{K. D. Elworthy} and \textit{S. Rosenberg}, Acta Appl. Math. 12, No. 1, 1--33 (1988; Zbl 0677.58046)