Singular sets of a class of fully non-linear equations in conformal geometry.

摘要

I am interested in the relations between fully non-linear elliptic equations and conformal geometry. For this, the σ_k curvatures of a locally conformally flat manifold; they constitute a natural generalization of the scalar curvature, and contain a lot of topological information about the manifold. From the analysis point of view, we study singularities of a fully non-linear elliptic PDE of Hessian type. Chapter 1 contains some motivation for this problem, and then some background can be found in Chapter 2.; The main ingredient is the study of the ‘almost’ divergence structure of σ_k together with an iteration process, those are described in Chapter 3.; In Chapter 4 we prove an estimate for the Hausdorff dimension of the singular set of a complete metric with positive σ_k curvature along with some topological consequences. Chapter 5 deals with isolated (and non isolated) singularities of the equation σ_k( v) = 1 and in particular we give a sufficient condition for removability of singularities. In chapter 6 we obtain a complete classification of point singularities for the subcritical problem without using additional hypothesis.