Partial regularity for homogeneous complex Monge-Ampère equations
In this note the authors briefly discuss how to prove the existence of almost smooth solutions of the following Dirichlet problem for the homogeneous complex Monge-Ampère equation: \[ (\pi^*_2\omega+ \partial\overline\partial \varphi)^{n+1}= 0\quad\text{on }\Sigma\times M,\tag{1} \] \[ \varphi|_{\partial\Sigma\times M}= \psi,\tag{2} \] where \(\Sigma\) is the unit disc in \(\mathbb C\). The authors present a new partial regularity theory for (1)--(2), which is obtained by studying foliations by holomorphic disks and their relation to (1)--(2). This note is a sequel to the authors' former paper [C. R., Math., Acad. Sci. Paris 340, No. 4, 287--290 (2005; Zbl 1069.32017)]. The results, announced in this paper and the cited one, will appear with details in the authors' preprint `Geometry of Kähler metrics and foliations by holomorphic discs' (2004), see arXiv:math/0507148.
scientific article; zbMATH DE number 2153798