You can get a course announcement here: in either postscript or pdf
format.
You might like a list of references (especially as there will be no
text book). Here's a start:
- Background (all books):
- Riemannian Manifolds: an Introduction to Curvature by J. M. Lee
(This is a great introduction to Riemannian geomtery)
- Riemannian Geometry: a Modern Introduction by I. Chavel (This
book has the best introduction to comparision theorems I've seen. Some people
tell me they have difficulty starting with it, though.)
- A Comprehensive Guide to Differential Geometry by M. Spivak
(particularly volumes II and IV. Ok, Spivak is notoriously
long-winded, but I really like his writing. And it is actually a close
to complete treatment of differential geometry.)
- Lectures in Differential Geometry by R. Schoen and S. T. Yau
(Just all the basics you need to know in Riemannian geometry,
presented by the masters. Curiously, this book was availible in
Chinese for a long time before it was availible in English.)
- Morse Theory by J. Milnor (The 20 or so page introduction to
Riemannian geometry is actually very good, if all you want is a quick
overview and/or refresher.)
- Elliptic Partial Differential Equations of Second Order by
D. Gilbarg and N. Trudinger (I'm going to just quote the appropriate
theorems from PDE; you can look up their proofs in this book
yourselves.)
- Some Nonlinear Problems in Riemannian Geometry by T. Aubin (This
book is incredibly dense, but it gives a fairly complete treatment of the
Yamabe problem and has very good references.)
- Survey Article: The Yamabe Problem by J. M Lee and T. Parker,
Bull Am. Math. Soc. Vol. 17 #1, 1987
- Closed Manifolds:
- On a Deformation of Riemannian Structures of Compact Manifolds by
H. Yamabe, Osaka Math. J. Vol. 12, 1960
- Remarks Concerning the Conformal Deformation of Riemannian
Structures on Compact Manifolds. by N. Trudinger, Ann. Scuola
Norm. Sup. Pisa Vol. 22, 1968
- Problemes Isoperimetriques et Espaces de Sobelev by T. Aubin,
J. Diff. Geom. Vol. 11, 1976
- Equations Differentielles Nonlineairers et Probleme de Yamabe
Concernant la Courbure Scalaire by T. Aubin, J. Math. Pures
Appl. Vol. 55, 1976
- Conformal Deformation of a Riemannian Metric to Constant Scalar
Curvature by R. Schoen, J. Diff. Geom. Vol. 20, 1984
- Variational Theory for the Total Scalar Curvature Functional for
Riemannian Metrics and Related Topics by R. Schoen, in Topics in
Calculas of Variations, ed. M. Giaquinta, 1987
- On the Number of Constant Scalar Curvature Metrics in a Conformal
Class by R. Schoen, in Differential Geometry for Manfredo Do Carmo,
ed. B. Lawson and K. Tenenblat, 1991
- Compact Manifolds with boundary: The Yamabe Problem on Manifolds
with Boundary by J. Escobar, J. Diff. Geom. Vol. 35 1992
- Noncompact, complete manifold without boundary:
- Conformally Flat Manifolds, Kleinian Groups and Scalar Curvature
by R. Schoen and S. T. Yau, Invent. Math. Vol. 92, 1988
- The Existence of Weak Solutions with Prescribed Singular Behavior
for a Conformally Invariant Scalar Equation by R. Schoen, Comm. Pure
Appl. Math Vol. 61, 1988
- Complete Conformal Metrics with Negative Scalar Curvature in
Compact Riemannina Manifolds by P. Aviles and R. McOwen, Duke
Math. J. Vol. 56, 1988
- Regularity for the Singular Yamabe Equation by R. Mazzeo,
Ind. Univ. Math. J. Vol. 40, 1991
- The Moduli Space of Singular Yamabe Metrics by R. Mazzeo,
D. Pollack and K. Uhlenbeck, J. Amer. Math. Soc. Vol. 9, 1996
Here are some references for analysis on noncompact manifolds,
either in postscript format or in pdf format.