Extension of the Hodge theorem to certain non-compact manifolds

We prove an analogue of the Hodge cohomology theorem for a certain class of non-compact manifolds. Specifically, let M be a compact manifold with boundary OM, and let g be a metric on Int(M). Assume that there exists a collar neighborhood of the boundary ... We then describe doubly weighted Sobolev spaces on M. For elements of these spaces the harmonic parts of w1 and w2 lie in one Sobolev space, while the non-harmonic parts of w1 and w2 lie in a differently defined Sobolev space. We prove that ... is Fredholm on almost all of these doubly weighted spaces, except for a finite number of values of w. This gives us an analogue of the Hodge decomposition theorem and leads to the result. This work generalizes earlier theorems of Atiyah, Patodi and Singer for b-metrics (case a = b = 0) and of Melrose for scattering metrics (case a = b = 1).

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2007

Author dc:creator

• Shapiro, Yakov (Yakov Mikhaylovich)

Advisor dc:contributor.advisor

• Richard B. Melrose.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/41724

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/41724