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This article is cited in 1 scientific paper (total in 1 paper)
Discs and boundary uniqueness for psh functions on almost complex manifold
A. Sukhovab a Université des Sciences et Technologies de Lille, Laboratoire Paul Painlevé, U.F.R. de Mathématiques, 59655 Villeneuve d'Ascq, Cedex, France
b Institute of Mathematics, Ufa Federal Research Center, RAS, 450008, Ufa, Russia
Abstract:
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.
Keywords:
almost complex structure, plurisubharmonic function, complex disc, totally real manifold.
Received: 19.06.2018
Citation:
A. Sukhov, “Discs and boundary uniqueness for psh functions on almost complex manifold”, Ufa Math. J., 10:4 (2018), 129–136
Linking options:
https://www.mathnet.ru/eng/ufa455https://doi.org/10.13108/2018-10-4-129 https://www.mathnet.ru/eng/ufa/v10/i4/p130
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Abstract page: | 247 | Russian version PDF: | 100 | English version PDF: | 34 | References: | 62 |
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