A priori estimates for a generalised Monge-Ampère PDE on some compact Kähler manifolds

Abstract

We study a fully nonlinear PDE involving a linear combination of symmetric polynomials of the Kähler form on a Kähler manifold. A $C^0$ \emph{a priori} estimate is proven in general and a gradient estimate is proven in certain cases. Independently, we also provide a method-of-continuity proof via a path of Kähler metrics to recover the existence of solutions in some of the known cases. Known results are then applied to an analytic problem arising from Chern-Weil theory and to a special Lagrangian-type equation arising from mirror symmetry.

Final version. To appear in Complex Variables and Elliptic Equations

References in corpus (2)

• The degenerate special Lagrangian equation
• Generalized complex Monge-Ampère type equations on closed Hermitian manifolds

Cited by in corpus (1)

• (1,1) forms with specified Lagrangian phase: A priori estimates and algebraic obstructions