Supported blow-up and prescribed scalar curvature on \(S^n\). (English) Zbl 1226.53040
Mem. Am. Math. Soc. 1002, v, 99 p. (2011).
This memoir consists of nine chapters. It deals with the problem of finding conditions sufficient for a given scalar function on a unit sphere \(S^n\) to be the curvature of a conformal metric on \(S^n\). There is a large bibliography of the subject.
Chapter 1, Introduction, contains a discussion of the problem and the statement of the main theorem. The remaining eight chapters build up the proof of the main theorem.
The formulation of the main theorem requires quite a few definitions and for this reason is not given here. The author uses blow-up sequences, that he calls supported, for a proof of the existence of positive \(C^2\) solutions to the basic (known) nonlinear partial differential equation, to which the problem is reduced.
Chapter 1, Introduction, contains a discussion of the problem and the statement of the main theorem. The remaining eight chapters build up the proof of the main theorem.
The formulation of the main theorem requires quite a few definitions and for this reason is not given here. The author uses blow-up sequences, that he calls supported, for a proof of the existence of positive \(C^2\) solutions to the basic (known) nonlinear partial differential equation, to which the problem is reduced.
Reviewer: Alexander G. Ramm (Manhattan/KS)
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
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