Elsevier

Advances in Mathematics

Volume 370, 26 August 2020, 107237
Advances in Mathematics

The C0 estimate for the quaternionic Calabi conjecture

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Abstract

We prove the C0 estimate for the quaternionic Monge-Ampère equation on compact hyperKähler with torsion manifolds. Our goal is to provide a simpler proof than the one presented in [3].

Section snippets

Introduction and preliminaries

The subject of this note is the quaternionic Monge-Ampère equation on a compact hyperKähler with torsion, later abbreviated as HKT, manifold.

We start by briefly reminding what are HKT manifolds. Those belong to the realm of quaternionic geometries and emerged from mathematical physics as the internal space of certain super-symmetric sigma models. The established reference for a mathematical treatment is [9] which we follow below. Let us recall that a hypercomplex manifold is one, say M,

The C0 estimate for the equation (1.1)

In this note we apply the convention that unless explicitly stated any constant, not written of what it is dependent, is independent of ϕ. When we want to express on what the constant is dependent we put those quantities in brackets for example C(p,fL(M)). The same letter may denote different constants from line to line just to avoid unnecessary indexing. All the Lq norms are taken w.r.t the volume element (ΩΩ)n.

In Subsection 2.1 we prove the Cherrier type inequality, cf. (22) in [7] or

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