A gradient estimate in the Calabi-Yau theorem
This paper is a nice contribution to the theory of complex Monge-Ampère equations on compact Kähler manifolds. The setting is the same as in Yau's proof of the Calabi conjecture [\textit{S.-T. Yau}, Commun. Pure Appl. Math. 31, 339--411 (1978; Zbl 0369.53059)]: \((M,\omega)\) is a compact Kähler manifold and \(\varphi\) is a real function so that \(\omega+i\partial\overline{\partial}\varphi\) is positive definite and with \(\max_M \varphi=0\). One assumes that \[ (\omega+i\partial\overline{\partial}\varphi)^n=f \omega^n, \] for some positive smooth function \(f\) with \(\int_M (f-1)\omega^n=0\), which is the complex Monge-Ampère equation that arises in the Calabi conjecture. In his solution of the conjecture, Yau proved that the function \(\varphi\) has \textit{a priori} bounds in all \(C^k\) norms, depending only on \(M, \omega, f\). To show this, he first proved that the \(C^0\) norm of \(\varphi\) is bounded uniformly, and then used this to derive a \(C^2\) uniform bound. Standard interpolation estimates then provide a \(C^1\) bound on \(\varphi\), but the question remained whether one could derive a \(C^1\) bound on \(\varphi\) depending only on the \(C^0\) bound. The author previously worked on this problem in [Math. Z. 244, No. 1, 153--161 (2003; Zbl 1076.32036)] where he proved such a bound if \((M,\omega)\) has nonnegative bisectional curvature (a rather stringent hypothesis). In this paper, the author generalizes his previous work and proves the desired \(C^1\) bound on \(\varphi\), which depends only on \(M, \omega, f\) and a bound on \(\max_M |\varphi|\). This result has already had some interesting applications, for example by the author as well as D. H. Phong and J. Sturm to the problem of geodesics in the space of Kähler potentials. The author promises further applications to complex Hessian equations.
scientific article; zbMATH DE number 5547449
scientific article