12/3/2004

Matt Gursky
University of Notre Dame

Prescribing symmetric functions of the eigenvalues of the Ricci tensor

In joint work with J. Viaclovsky, we studied the problem of prescribing symmetric functions of the eigenvalues of the Schouten tensor for a conformal metric on a compact manifold (often referred to as the "Sigma-k Yamabe problem").  This is equivalent to solving a fully nonlinear elliptic equation of second order.  Assuming the function satisfies certain structural conditions, and the underlying manifold satisfies a natural 'admissibility' condition, we prove a priori estimates for solutions.  The proof involves a blow-up analysis and classification of certain global singular solutions.