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
URL:
http://dx.doi.org/10.1007/BF00047567
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