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

TL;DR

This paper establishes a sharp uniform $C^0$ estimate for the quaternionic Monge-Ampere equation on hyperhermitian manifolds, improving previous bounds and extending applicability to various initial metrics and related quaternionic PDEs.

Contribution

It provides the first sharp $C^0$ estimate depending only on $L^p$ norms for $p>2$, applicable to all hyperhermitian metrics, and extends the estimate to related quaternionic PDEs.

Findings

The estimate depends solely on the $L^p$ norm of the right-hand side for $p>2.

The estimate holds for any hyperhermitian initial metric, not just HKT.

A unified sharp uniform estimate is provided for various quaternionic PDEs.

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].