You can get a course announcement here: in either postscript or pdf format.

• 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.