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

DOI10.1007/S00526-024-02694-9arXiv2211.00959OpenAlexW4394785817MaRDI QIDQ6415894

Publication date: 2 November 2022

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].

Full work available at URL: https://doi.org/10.1007/s00526-024-02694-9

Mathematics Subject Classification ID

Global differential geometry of Hermitian and Kählerian manifolds (53C55) A priori estimates in context of PDEs (35B45) Hyper-Kähler and quaternionic Kähler geometry, ``special geometry (53C26) Monge-Ampère equations (35J96) PDEs on manifolds (35R01)