Non-linear Partial Differential Equations and Applications to Problems in Geometry

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PI: Sun-Yung Alice Chang/ Paul Yang, Princeton UniversityDMS-0245266Abstract:The analytic part of the proposal is concerned with two questions concerning nonlinear differential equations arising from conformal geometry. The first is to formulate a general notion of weak solutions for a family of fully nonlinear equations containing the equations to prescribe the symmetric functions of the Weyl-Schouten tensor, and to provide criteria for removal of singularities of such equation. This would open the way to find solutions to these equations by a more traditional variational method. The second is to find Sobolev inequalities for fourth order equations that arise in conformal geometry in dimensions three and four. The geometric part of this proposal is concerned with applications of our recent work in prescribing the second symmetric functions of theWeyl-Schouten tensor on a four-dimensional manifold to the diffeomorphisms classification of a class of four-dimensional manifolds, as well as applications to the study of Kleinian groups in higher dimensional manifolds.This proposal is concerned with new methods to solve a family of nonlinear differential equations that is associated with conformal geometry in which the primary data is the knowledge of angle measurements. The ability to solve these equations gives us new numerical invariants that will eventually allow us to classify these geometries in three and four dimensions a subject that is of wide interest in the geometry and topology community. The differential equations are highly nonlinear and appear among a family of such equations that have only recently yielded to a geometric approach. Such development will generate a set of new tools to analyze the structure nonlinear partial differential equations a subject also of wide interest in general.