Complex Hessian measures with respect to a background Hermitian form

Abstract

We develop potential theory for $m$-subharmonic functions with respect to a Hermitian metric on a Hermitian manifold. First, we show that the complex Hessian operator is well-defined for bounded functions in this class. This allows to define the $m$-capacity and then showing the quasi-continuity of $m$-subharmonic functions. Thanks to this we derive other results parallel to those in pluripotential theory such as the equivalence between polar sets and negligible sets. The theory is then used to study the complex Hessian equation on compact Hermitian manifold with boundary, with the right hand side of the equation admitting a bounded subsolution. This is an extension of a recent result of Collins and Picard dealing with classical solutions.

57 pages, V3 with some minor corrections. This is the final version, to appear in APDE

References in corpus (7)

• The complex Monge-Ampère equation on compact Hermitian manifolds
• A priori estimates for the complex Hessian equations
• Gauduchon metrics with prescribed volume form
• Viscosity solutions to complex Hessian equations
• Hessian equations on closed Hermitian manifolds
• Weak solutions of complex Hessian equations on compact Hermitian manifolds
• A Wess--Zumino--Witten type equation in the space of Kähler potentials in terms of Hermitian--Yang--Mills metrics