The formation of singularities in the harmonic map heat flow (Q678097)

Let \(M\) and \(N\) be compact Riemannian manifolds. This paper deals with a flow of maps from \(M\) to \(N\) given by the harmonic map heat flow \(\partial F/\partial t = \Delta F\). Assume \(F\) be a solution for a finite time interval \([0,T)\) with bounded energy \(E\). Then the authors first prove that there exists an exceptional set \(S\) of finite \(m-2\) dimensional Hausdorff measure in \(M\), where \(m\) is the dimension of \(M\), out of which the flow smoothly converges as \(t\rightarrow T\). The \(m-2\) dimensional Hausdorff measure of \(S\) is bounded in terms of \(E\) and certain Dirichlet integral of \(F\). Second, they examine how singularity forms inside \(S\). Assuming the singularity forms rapidly, by rescaling the time at \(T\) they prove the existence of infinitesimal shape of the limiting maps, i.e. the limit flow defined on the tangent space at singular point to the target manifold \(N\). The obtained flow is dilation-invariant and proved to be nonconstant. This paper contains not a few typos and, though the reviewer cannot guess why the error has occured, the graphs mentioned in p. 528 are not included.