A study of regularity problem of harmonic maps (Q1118875)

Let M and N be closed compact Riemannian manifolds with dimensions dim \(M\geq 3\) and \(u\in L^ 2_ 1(M,N)\). Consider \(u_ t=u\cdot \phi_ t\) for \(\phi_ t\) a 1-parameter family of compactly supported \(C^ 1\) diffeomorphisms of M with \(\phi_ 0=id\). If u is a critical point of the energy functional for all variations of this type and if u is harmonic then u is called a stationary map. In this paper, the author studies stationary maps whose singular set is of codimension greater than 2. The main result is as follows: Suppose u is a stationary map, whose singular set is contained in the graph of a \(C^{1,\alpha}\) function with dimension \(d<n-2\). There exists an \(\epsilon >0\) such that u is regular if the energy E(u)\(\leq \epsilon\).