Regularity of harmonic maps with prescribed singularities
Let \(M\) be a complete Riemannian manifold and \(N\) a codimension two closed submanifold. Given a smooth map \(h: M\setminus N\to \overline{H^ m}\) into the naturally compactified hyperbolic space \(H^ m\), the authors show the existence of a harmonic map \(\varphi: M\setminus N\to H^ m\) with similar asymptotic behaviour to \(h\) along \(N\). Here, `similar behavior' means that, in terms of the coordinate functions \(h_ 1,\dots,h_ m\) of \(h\) in the upper half-space model of \(H^ m\), \(\varphi\) is of the form \((\varphi_ 1,\dots,\varphi_{m-1},h_ me^{\varphi_ m})\), with \(\varphi_ i-h_ i\) belonging to a suitable completion of \(C^ \infty_ 0(M)\) or \(C^ \infty_ 0(M\setminus N)\). This is then used to give a partial solution to the Hawking's conjecture to the effect that there is no asymptotically flat axially symmetric solution to the Einstein Vacuum Equation with disconnected event horizon of small angular momentum.
scientific article; zbMATH DE number 95693