Ufa Mathematical Journal
    Volume 10, Number 4, pp. 129-136

    Discs and boundary uniqueness for psh functions on almost complex manifold

    Sukhov A.B.


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    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.