Function theory and Sobolev inequalities have been the target of investigation 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 and describe new results.
Euclidean background Statement of the $AB$ program Some historical motivations The $H^2 1$-inequality--Part I The $H^2 1$-inequality--Part II PDE methods The isoperimetric inequality The $H^p 1$-inequalities, $1 < p < dimM$ Bibliography.