Memoirs of the American Mathematical Society 2002; 98 pp; softcover Volume: 160 ISBN10: 0821829890 ISBN13: 9780821829899 List Price: US$59 Individual Members: US$35.40 Institutional Members: US$47.20 Order Code: MEMO/160/761
 Function theory and Sobolev inequalities have been the target of investigatio for decades. Sharp constants in these inequalities constitute a critical tool in geometric analysis. The \(AB\) program is concerned with sharp Sobolev inequalities on compact Riemannian manifolds. Important and significant progress has been made during recent years. We summarize the present state ad describe new results. Readership Graduate students and research mathematicians interested in global analysis and analysis on manifolds. Table of Contents  Euclidean background
 Statement of the \(AB\) program
 Some historical motivations
 The \(H^2_1\)inequalityPart I
 The \(H^2_1\)inequalityPart II
 PDE methods
 The isoperimetric inequality
 The \(H^p_1\)inequalities, \(1 < p < \mathrm{dim}M\)
 Bibliography
