2026-09-20T08:53:12Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/417242022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Richard B. Melrose.
Shapiro, Yakov (Yakov Mikhaylovich)
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2008-05-19T16:12:19Z
2008-05-19T16:12:19Z
2007
2007
http://hdl.handle.net/1721.1/41724
225064035
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007.
Includes bibliographical references (p. 92).
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).
by Yakov Shapiro.
Ph.D.
92 p.
eng
Massachusetts Institute of Technology
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Extension of the Hodge theorem to certain non-compact manifolds
Thesis
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