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On \(L^\infty\) estimates for complex Monge-Ampère equations. (English) Zbl 1525.35121

\(L^{\infty} \) estimates for the complex Monge-Ampère equations on compact Kähler manifolds, via Moser iteration, were one of the crucial points in Yau’s proof of the Calabi conjecture. Later the reviewer proposed a different method, based on properties of positive currents, which works for nonnegative right hand side with density in \(L^p\) , \(p>1\). In the present paper, the authors introduce a new approach, using PDEs techniques, which allows to prove the latter result and which is applicable in more general settings.
Let \((X, \omega _X )\) be a compact \(n\)-dimensional Kähler manifold. For a smooth function \(u\) the notation \(\omega _u =\omega _X +i\partial\bar{\partial} u \) is used. Fix local coordinates and compose the matrix of coefficients of \(\omega _u\) with the inverse matrix of \(\omega _X\) to define an endomorphism \(h_u\). Denoting by \(\lambda [h_u]\) the vector of eigenvalues of that endomorphism we consider the equation \[ f ( \lambda [h_u ]) =ce^F, \tag{\(\ast\)}\] where \(c>0\) is a constant (depending on the equation), \(F\) a suitably normalized function, and \(\lambda [h_u] \in \Gamma\) with \(\Gamma\) a cone with vertex at \(0\), contained in the set where the sum of \(\lambda _j\) is positive and containing the first (positive) octant. The operator \(f\) is elliptic (increasing in each \(\lambda _j\)), homogeneous of order \(1\), and it satisfies for some positive \(\gamma\) \[ det \left( \frac{ \partial f ( \lambda [h])}{\partial h_{jk} }\right) >\gamma . \] The given function \(F\) is assumed to have finite \(p\)-entropy \[ \int _X e^{nF}|F|^p \omega _X ^n <\infty \] for \(p>n\). The main theorem says that with the above hypothesis any \(C^2\) solution \(u\) of equation \((*)\) satisfies \[ \sup |u| \leq C , \] where \(C\) depends on \(n,p,\gamma, c, \omega _X \), the volume of \(X\) and the \(p\)-entropy.
The theorem covers the complex Monge-Ampère equation, the complex Hessian equation and the complex quotient Hessian equation. In a second statement it is extended so that the estimete holds (uniformly) for families of metrics approximating a semipositive form, which is no longer Kähler.
The new idea of the proof is the use of the solution \(v\) of an auxiliary complex Monge-Ampère equation with the right hand depending on \(u\) which solves \((*)\). Then the barrier function involving \(u\) and \(v\) is cleverly chosen and it is shown that under the assumptions it must be negative. This finally leads, via a De Giorgi-type lemma, to the estimate on \(u\).

MSC:

35J60 Nonlinear elliptic equations
35J96 Monge-Ampère equations
32W20 Complex Monge-Ampère operators
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C56 Other complex differential geometry
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