Compactness and Non-compactness for the Yamabe Problem on Manifolds With Boundary
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Authors
Disconzi, Marcelo Mendes
Issue Date
1-May-12
Type
Dissertation
Language
en_US
Keywords
Alternative Title
Abstract
We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension $n \leq 24$. The Weyl Vanishing Theorem is also established under these hypotheses, and we provide counter-examples to compactness when $n \geq 25$. Lastly, our methods point towards a vanishing theorem for the umbilicity tensor, which is anticipated to be fundamental for a study of the nonumbilic case.
Description
116 pg.
Citation
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.