Fourth Order Equations in Conformal Geometry

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ABSTRACTThe main theme of the project is to develop the theory of fourth orderequations arising from conformal geometric considerations. The firstpart of the project seeks to apply the fourth order equation as regularizing equations of the more natural but more difficult fullynonlinear partial differential equations in conformal geometry.The first consequence will be natural conformal criteria for theexistence of a conformal metric of positive Ricci curvature. The secondpart of the project seeks to widen the set of conformal classes to whichour theory of fourth order equations may be applied. A concrete problemwould be to find saddle point solutions of the fourth order equationson a large class of four dimensional conformal structures of negativescalar curvature. The third part of the proposed project is to developthe theory of biharmonic maps in four dimension. It is proposed todevelop a classification of biharmonic maps of special conformalstructures. The several parts of this proposal share a common novel feature toconsider a fourth order nonlinear differential equation that relatesfourth order term of a scalar quantity to the scalar quantity itself.The equation came from geometric considerations for a four dimensionalspace that are natural from topological perspective. The projectwill have an impact on the development of higher order differentialequations in several ways. First it will introduce new techniquesto study the solutions of such equations. Secondly, it will yield newmethods to solve the more traditional second order equations of geometricinterest. Thirdly, the results have aleady yield application to thegeometry of four dimensional space.