The Monge-Ampère equation for (n − 1)-plurisubharmonic functions on a compact Kähler manifold

By Valentino Tosatti, B. Weinkove. 2013

139 citations recorded by the source corpus.

A C function on C is called (n− 1)-plurisubharmonic in the sense of Harvey-Lawson if the sum of any n − 1 eigenvalues of its complex Hessian is nonnegative. We show that the associated MongeAmpère equation can be solved on any compact Kähler manifold. As a consequence we prove the existence of solutions to an equation of FuWang-Wu, giving Calabi-Yau theorems for balanced, Gauduchon and strongly Gauduchon metrics on compact Kähler manifolds.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0bfed3eb9e9a4c35ab45568168c91045ebd4d32a

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The Monge–Ampère equation for non-integrable almost complex structures (2016, background): “1) is replaced by a Monge-Ampère type equation for (n − 1)-plurisubharmonic functions [26, 17, 18, 38, 50, 51].”
Fully non-linear parabolic equations on compact Hermitian manifolds (2017, unknown): “As the argument in [50, 57], it follows from (2.”
Gauduchon metrics with prescribed volume form (2015, methodology): “ more generally if the linear term involving ∂uis removed, the equation (1.7) reduces to the Monge-Amp`ere equation for (n− 1)-plurisubharmonic functions, solved by the second and third-named authors [39, 40] (see also [11] for earlier partial results). (2) In the case when n= 2 this equation reduce”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, background): “ and ωn−1 0 + √ −1∂∂(uωn−2) is a positive (n−1,n−1) form. This choice of χgives an equation introduced by Fu-Wang-Wu [25]. It is not known in general whether it can be solved (except when ωis Ka¨hler [60].) The C2,α estimates for (1.6) in the above settings (again, depending on two derivatives of ψ)”