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  • 1
    ISSN: 1572-9036
    Keywords: 58G11 ; 58G32 ; 60H15 ; Bochner's theorem ; Brownian motion and heat flow ; spectrum of the Laplacian ; hyperbolic manifolds ; cohomology with compact support
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Bochner's theorem that a compact Riemannian manifold with positive Ricci curvature has vanishing first cohomology group has various extensions to complete noncompact manifolds with Ricci possibly negative. One still has a vanishing theorem for L 2 harmonic one-forms if the infimum of the spectrum of the Laplacian on functions is greater than minus the infimum of the Ricci curvature. This result and its analogues for p-forms yield vanishing results for certain infinite volume hyperbolic manifolds. This spectral condition also imposes topological restrictions on the ends of the manifold. More refined results are obtained by taking a certain Brownian motion average of the Ricci curvature; if this average is positive, one has a vanishing theorem for the first cohomology group with compact supports on the universal cover of a compact manifold. There are corresponding results for L 2 harmonic spinors on spin manifolds.
    Type of Medium: Electronic Resource
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