Non-linear Partial Differential Equations in Conformal Geometry

In the study of conformal geometry, the method of elliptic partial differential equations is playing an increasingly significant role. Since the solution of the Yamabe problem, a family of conformally covariant operators (for definition, see section 2) generalizing the conformal Laplacian, and their associated conformal invariants have been introduced. The conformally covariant powers of the Laplacian form a family P2k with k ∈ N and k ≤ n 2 if the dimension n is even. Each P2k has leading order term (−∆) and is equal to (−∆) if the metric is flat. The curvature equations associated with these P2k operators are of interest in themselves since they exhibit a large group of symmetries. The analysis of these equations is of necessity more complicated, it typically requires the derivation of an optimal Sobolev or Moser-Trudinger inequality that always occur at a critical exponent. A common feature is the presence of blowup or bubbling associated to the noncompactness of the conformal group. A number of techniques have been introduced to study the nature of blowup, resulting in a well developed technique to count the topological degree of such equations. The curvature invariants (called the Q-curvature) associated to such operators are also of higher order. However, some of the invariants are closely related with the Gauss-Bonnet-Chern integrand in even dimensions, hence of intrinsic interest to geometry. For example, in dimension four, the finiteness of the Q-curvature integral can be used to conclude finiteness of topology. In addition, the symmetric functions of the Ricci tensor appear in natural fashion as the lowest order terms of these curvature invariants, these equations offer the possibility to analyze the Ricci tensor itself. In particular, in dimension four the sign of the Q-curvature integral can be used to conclude the sign of the Ricci tensor. Therefore there is ample motivation for the study of such equations.