A second order estimate for complex Hessian equations on a compact Kähler manifold

On the geometric flows solving Kählerian inverseσkequations

“…Among them, one important example is Yau's seminal work on the complex Monge-Ampère equations in the Calabi conjecture. The general case has been studied recently in [Hou 2009;Hou et al 2010]. There exist, however, some analytical difficulties in completely solving this problem for k < n.…”

Hölder Continuous Solutions to Complex Hessian Equations

“…A major progress has been done recently both for domains in C n (see [2,9,20]), and on compact Kähler manifolds, (see [10,17]). In particular, the Calabi-Yau type theorem for complex Hessian equations on a compact Kähler manifold was proved in [10].…”

Solutions to degenerate complex Hessian equations

“…The non degenerate complex Hessian equation on compact Kähler manifold, where F (x, ϕ) = f (x), with 0 < f ∈ C ∞ (X), has been studied recently in [H09,HMW10,Jb10,DK12]. In [H09] and [Jb10], the authors independently solved this equation with a strong additional hypothesis, assuming (X, ω) has non negative holomorphic bisectional curvature.…”

“…In [H09] and [Jb10], the authors independently solved this equation with a strong additional hypothesis, assuming (X, ω) has non negative holomorphic bisectional curvature. Later on, in [HMW10] an a priori C 2 estimate was obtained without curvature assumption. Recently, using this estimate and a blowing up analysis suggested in [HMW10], Dinew and Kolodziej solved the equation in full generality.…”

“…Later on, in [HMW10] an a priori C 2 estimate was obtained without curvature assumption. Recently, using this estimate and a blowing up analysis suggested in [HMW10], Dinew and Kolodziej solved the equation in full generality.…”