P-harmonic morphisms, minimal foliations, and conformal deformations of metrics /

Abstract

In this dissertation, we study p-harmonic morphisms and its interaction with minimal foliations and conformal deformations of metrics. We give several methods to construct non-trivial p-harmonic morphisms via conformal deformations of metric on the domain and/or target manifold. We classify polynomial p-harmonic morphisms between Euclidean spaces and holomorphic p-harmonic morphisms between complex Euclidean spaces. We find three applications of p-harmonic morphisms including applications to the study of biharmonic morphisms and in showing the existence of harmonic 3-sphere in a general Riemannian manifold with noncontractible universal covering space. Finally, we give links between p-harmonicity of functions and the minimality of their level hypersurfaces or of their vertical graphs. We prove that the foliation defined by the level hypersurfaces of a submersive p-harmonic function or by the vertical graphs of a harmonic function can always be turned into a minimal foliation via a suitable conformal deformation of metric.