On Constructing Biharmonic Maps and Metrics

We construct biharmonic nonharmonic maps between Riemannian manifoldsM and N by first making the ansatz that M N be aharmonic map and then deforming the metric conformally on M to render biharmonic. The deformation will, in general, destroy theharmonicity of . We call a metric which renders the identity mapbiharmonic, a biharmonic metric. On an Einstein manifold, theonly conformally equivalent biharmonic metrics are defined byisoparametric functions.     

Unique continuation of semi-conformality for a harmonic mapping onto a surface

We show that a harmonic mapping ϕ from either a three-manifold (with a condition on its Ricci curvature) or from a surface with values in a surface which has rank 2 somewhere, satisfies the following unique continuation property: if ϕ is semi-conformal on an open set, then it is semi-conformal everywhere.