Uniqueness Of Tangent Cone For Biharmonic Maps And Expansions Of Ricci Flat ALE Metrics

This dissertation consists of two parts.In the first part,we study the problem of uniqueness of tangent cone for minimizing extrinsic biharmonic maps.We prove that if the target manifold is a compact analytic submanifold in RP and if there is one tangent map whose singularity set consists of the origin only,then this tangent map is unique.In the second part,we study the full regularity of a class of Ricci flat metrics in the harmonic coordinates about the infinity of ALE(asymptotically local Euclidean)manifolds,in particular we get the expansions up to any order of the metric coefficients.