Bergman-harmonic maps of balls (Q2807100)

Let \((\Omega ,g)\) and \((D,h)\) denote Kähler manifolds and \(\Phi:\Omega \longrightarrow D\) be a holomorphic map. Then, if \(\Omega\) and \(D\) are compact, a result of A. Lichnerowicz yields that \(\Phi\) is a stable harmonic map and an absolute minimum in its homotopy class for the Dirichlet energy functional. If \(\Omega\) and \(D\) are allowed to be non-compact, this result fails. In the first part the authors consider the case that \(\Omega\) and \(D\) are balls and \(g\) is the Bergman metric, showing a suitably modified version of Lichnerowicz's result. In a second part Bergman-harmonic maps between balls \(\mathbb B^n\) and \(\mathbb B^N\) are studied that extend \(C^2\) to the boundary. It is shown that they have to satisfy a compatibility system similar to the tangential Cauchy-Riemann equations. A third main theorem treats weakly differentiable Bergman-harmonic maps \(\Phi\) between \((\mathbb B^n, g)\) and \((\mathbb B^N,h)\) admitting \(L^2\)-boundary values \(\phi\). It is shown that, under reasonable conditions, \(\phi\) is a weakly subelliptic harmonic map.