The Profile of Bubbling Solutions of a Class of Fourth Order Geometric Equations on 4-manifolds

Let (M,g) be a compact Riemannian manifold. The conformal class of g consists of all metrics g̃ = e2ug for any smooth function u. A central theme in conformal geometry is the study of properties that are common to all metrics in the same conformal class, and the understanding and classification of all the conformal classes. For this purpose it is often useful to be able to single out a unique representative in each conformal class by imposing some geometric condition. This usually leads to a conformally covariant geometric equation for the conformal factor e2u. Such equations have attracted much interest in the literature in the past half-century. In dimension 2, the natural condition to impose is constant Gauss curvature. The Poincaré Uniformization Theorem states that this is always possible: every compact Riemannian surface is conformal to one with constant Gauss curvature. The conformally covariant operator in this case is the Laplacian-Beltrami operator, given in local coordinates by: