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A Subsolution Theorem for the Monge-Ampère Equation over an Almost Hermitian Manifold

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Abstract

Let Ω ⊆ M be a bounded domain with a smooth boundary ∂Ω, where (M, J, g) is a compact, almost Hermitian manifold. The main result of this paper is to consider the Dirichlet problem for a complex Monge-Ampère equation on Ω. Under the existence of a C2-smooth strictly J-plurisubharmonic (J-psh for short) subsolution, we can solve this Dirichlet problem. Our method is based on the properties of subsolutions which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds.

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Acknowledgements

The author would like to thank his thesis advisor Professor ** Zhang for his constant support and advice.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

    Jiaogen Zhang  (张教根)

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  1. Jiaogen Zhang  (张教根)
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Correspondence to Jiaogen Zhang  (张教根).

Additional information

The research was supported by the National Key R and D Program of China (2020YFA0713100).

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Zhang, J. A Subsolution Theorem for the Monge-Ampère Equation over an Almost Hermitian Manifold. Acta Math Sci 42, 2040–2062 (2022). https://doi.org/10.1007/s10473-022-0518-9

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  • DOI: https://doi.org/10.1007/s10473-022-0518-9

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