Cetraro, June 11-76, 2007 PDEs in Conformal Geometry Prof. Matthew Gursky, Notre Dame Univ., USA Abstract These lectures are intended to provide an introduction to some important elliptic PDEs which arise in conformal geometry: the scalar curvature equation, the Q-curvature equation, the functional determinant, and the fully nonlinear Yamabe equation, with an emphasis on the last two problems. Lecture I Background in Riemannian geometry: curvature and its transformation under a conformal change of metric. The Bochner formula. Lecture II The Uniformization Theorem and the scalar curvature equation. Conformal transformations of the sphere; "bubbling", epsilon-regularity. The determinant of the laplacian on a surface. Lecture III Four dimensions: the log-det formulas of Branson-Orsted, Chang-Yang. The Q-curvature, the sigma-2 curvature. Lecture IV The functional determinant: some existence and regularity results. Lecture V Fully nonlinear equations in geometry. The sigma-k Yamabe problem. Local estimates and epsilon-regularity. Counterexamples. S.Y.-A. Chang, "Nonlinear Elliptic Equations in Conformal Geometry", Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2004. Peter Li, "Lecture Notes on Geometric Analysis." Lecture Notes Series No. 6, Research Institute of Mathematics, Global Analysis Research Center, Seoul National University, Korea (1993) Jeff Viaclovsky, "Conformal Geometry and Nonlinear Equations", to appear in World Scientific Memorial Volume for S.S. Chern, 2006. |