Richard Schoen's 1988 Graduate Course on Scalar Curvature
In Autumn, 1988, when I was beginning my third year in graduate school at Stanford University, Richard Schoen taught a wonderful topics course on Scalar Curvature. The notes that I wrote up from that course have been widely circulated and cited in a number of research publications over the last 25 years. In order to make these notes more easily accessible, and with Rick's permission, I am making a scanned copy of them available as a pdf here on this web site. If you choose to cite these notes please include a link to this url so that others may access them.
The later portion of the notes consist of a detailed sketch of a proof of the compactness for constant scalar curvature metrics within a conformal class on a compact manifold in low dimensions (less than or equal to 8). A few years ago, through combined work of Brendle, Khuri, Marques and Schoen, the general case was resolved: compactness holds in dimensions less than or equal to 24 and is false in higher dimensions. Below I have included references to some of the relevant papers in this work. This is an incomplete list and I urge the interested reader to consult the references in the papers below for a complete history.
Dan Pollack's course notes from Richard Schoen's 1988 graduate course:
Some relevant papers regarding compactness for the Yamabe problem:
Richard M. Schoen
Variational theory for the total scalar curvature functional for Riemannian metrics and related topics.
Topics in calculus of variations (Montecatini Terme, 1987), 120-154, Lecture Notes in Math., 1365, Springer, Berlin (1989).
Marcus Khuri, Fernando C. Marques, and Richard M. Schoen
A compactness theorem for the Yamabe problem.
J. Differential Geom. 81 (2009), no. 1, 143-196.
Fernando C. Marques
Blow-up examples for the Yamabe problem.
Calc. Var. Partial Differential Equations 36 (2009), no. 3, 377-397.
Simon Brendle and Fernando C. Marques
Blow-up phenomena for the Yamabe equation II.
J. Differential Geom. 81 (2009), no. 2, 225-250.
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