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Complex Monge–Ampère equation for measures supported on real submanifolds

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Abstract

Let \((X,\omega )\) be a compact n-dimensional Kähler manifold on which the integral of \(\omega ^n\) is 1. Let K be an immersed real \(\mathcal {C}^3\) submanifold of X such that the tangent space at any point of K is not contained in any complex hyperplane of the (real) tangent space at that point of X. Let \(\mu \) be a probability measure compactly supported on K with \(L^p\) density for some \(p>1\). We prove that the complex Monge–Ampère equation \((dd^c \varphi + \omega )^n=\mu \) has a Hölder continuous solution.

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Acknowledgements

The author would like to thank Tien-Cuong Dinh for introducing him this research topic and for his illuminating discussions. He also wants to express his gratitude to the anonymous referee for his useful remarks which improved considerably the presentation of the paper and to Lucas Kaufmann for fruitful discussions. “Funding was provided by Region Ile-de-France”.

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Authors and Affiliations

  1. UMR 7586, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UPMC Univ Paris 06, 4 place Jussieu, 75005, Paris, France

    Duc-Viet Vu

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  1. Duc-Viet Vu
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Correspondence to Duc-Viet Vu.

Additional information

Communicated by Ngaiming Mok.

This research is supported by grants from Région Ile-de-France.

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Vu, DV. Complex Monge–Ampère equation for measures supported on real submanifolds. Math. Ann. 372, 321–367 (2018). https://doi.org/10.1007/s00208-017-1565-8

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  • DOI: https://doi.org/10.1007/s00208-017-1565-8

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