Discs and boundary uniqueness for psh functions on almost complex manifold

Аннотация

This paper is inspired by the work by J.-P. Rosay (2010). In this work, there was sketched a proof of the fact that a totally real submanifold of dimension $2$ can not be a pluripolar subset of an almost complex manifold of complex dimension $2$. In the present paper we prove a considerably more general result, which can be viewed as a boundary uniqueness theorem for plurisubharmonic functions. It states that a function plurisubharmonic in a wedge with a generic totally real edge is equal to $-\infty$ identically if it tends to $-\infty$ approaching the edge. Our proof is completely different from the argument by J.-P. Rosay. We develop a method based on construction of a suitable family of $J$-complex discs. The origin of this approach is due to the well-known work by S. Pinchuk (1974), where the case of the standard complex structure was settled. The required family of complex discs is obtained as a solution to a suitable integral equation generalizing the classical Bishop method. In the almost complex case this equation arises from the Cauchy–Green type formula. We hope that the almost complex version of this construction presented here will have other applications.