Regularity of the degenerate Monge-Ampère equation on compact Kähler manifolds
Let \(M\) be a compact Kähler manifold of complex dimension \(n\geq 2\), with Kähler form \(\omega\). A well known result of \textit{S.-T.~Yau} [Commun. Pure Appl. Math. 31, 339--411 (1978; Zbl 0369.53059)] states that for any positive function \(f\in C^\infty(M)\) there is a unique solution \(\phi\in C^\infty(M)\) of the complex Monge-Ampère equation \[ (dd^c\phi+\omega)^n = f\omega^n \quad\text{on}\quad M, \qquad dd^c\phi+\omega\geq 0 \quad\text{on}\quad M, \qquad \int_M \phi\omega^n=0. \leqno(*) \] Here the author is interested in the degenerate case \(f\geq 0\). In this case it is known from results of \textit{S.~Kołodziej} [Acta Math. 180, 69--117 (1998; Zbl 0913.35043); Indiana Univ. Math. J. 52, 667--686 (2003; Zbl 1039.32050)] that if \(f\in L^q(M)\) for some \(q>1\), then there is a unique continuous generalized solution \(\phi\) of \((*)\). The main result of the paper is that if \(f^{1/(n-1)}\in C^{1,1}(M)\), then any generalized solution \(\phi\) of \((*)\) is almost \(C^{1,1}\) in the sense that \(\Delta\phi\) is bounded (this is equivalent to the boundedness of the mixed complex derivatives \(\partial^2\phi/\partial z_i\partial\bar z_j\)). A further result is that if \(M\) has nonnegative bisectional curvature and \(f^{1/n}\in C^{0,1}(M)\), then \(D\phi\) is bounded. The assumption on the bisectional curvature can be dropped if \(\| \phi\| _{L^\infty}\) is small enough.
scientific article; zbMATH DE number 1944236