## 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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### References:

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