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

Abstract

A C^2 function on C^n 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 Monge-Ampere equation can be solved on any compact Kahler manifold. As a consequence we prove the existence of solutions to an equation of Fu-Wang-Wu, giving Calabi-Yau theorems for balanced, Gauduchon and strongly Gauduchon metrics on compact Kahler manifolds.

40 pages, final version to appear in JAMS

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Cited by in corpus (28)

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• Stability and regularity of solutions of the Monge-Ampère equation on Hermitian manifolds
• On the Alesker-Verbitsky conjecture on hyperKähler manifolds
• On a class of Hessian type equations on Riemannian manifolds
• Transverse Fully Nonlinear Equations on Sasakian Manifolds and Applications
• The Dirichlet Problem of Fully Nonlinear Equations on Hermitian Manifolds
• Geometric flows and Strominger systems
• A parabolic Monge-Ampère type equation of Gauduchon metrics
• Curvature estimates for a class of Hessian type equations
• The parabolic Monge-Ampère equation on compact almost Hermitian manifolds
• Regularity of fully non-linear elliptic equations on Hermitian manifolds. II
• Long time existence of the (n-1)-plurisubharmonic flow
• Ricci Curvatures on Hermitian manifolds
• Fully non-linear degenerate elliptic equations in complex geometry
• Parabolic complex Monge-Ampere equations on compact Kahler manifolds
• Hadamard-type inequalities for $k$-positive matrices
• On canonical metrics of complex surfaces with split tangent and related geometric PDEs
• Monge-Ampère type equations on almost Hermitian manifolds
• Second order estimates for fully nonlinear elliptic equations with gradient terms on Hermitian manifolds
• The Monge-Ampère equation for strictly $(n-1)$-convex functions with Neumann condition
• Fully non-linear elliptic equations on compact almost Hermitian manifolds
• Local regularity for concave homogeneous complex degenerate elliptic equations comparable to the Monge-Ampère equation
• The Dirichlet problem for a class of degenerate fully nonlinear elliptic equations on Riemannian manifolds with mean concave boundary
• Blowing up Chern-Ricci flat balanced metrics
• A Cheng-Yau type estimate for the symplectic Calabi-Yau equation
• Positivity in Foliated Manifolds and Geometric Applications