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arXiv · 2211.00959

Sharp uniform bound for the quaternionic Monge-Ampere equation on hyperhermitian manifolds

Marcin Sroka
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Abstract

We provide the sharp $C^0$ estimate for the quaternionic Monge-Ampere equation on any hyperhermitian manifold. This improves previously known results concerning this estimate in two directions. Namely, it turns out that the estimate depends only on $L^p$ norm of the right hand side for any $p>2$ (as suggested by the local case studied in [Sr20a]). Moreover, the estimate still holds true for any hyperhermitian initial metric - regardless of it being HKT as in the original conjecture of Alesker-Verbitsky [AV10] - as speculated by the author in [Sr21]. For completeness, we actually provide a sharp uniform estimate for many quaternionic PDEs, in particular those given by the operator dominating the quaternionic Monge-Ampere operator, by applying the recent method of Guo and Phong [GP22a].

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Marcin Sroka. 2022-11-02. Sharp uniform bound for the quaternionic Monge-Ampere equation on hyperhermitian manifolds. https://doi.org/10.1007/s00526-024-02694-9

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