Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds

Abstract

A quaternionic version of the Calabi problem on the Monge-Ampère equation is introduced, namely a quaternionic Monge-Ampère equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercomplex manifold with holonomy in SL(n, H), uniqueness (up to a constant) of a solution is proven, as well as the zero order a priori estimate. The existence of a solution is conjectured, similar to the Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of the complex Hessian equation, making sense on any complex manifold.

The continuity and range of the quaternionic Monge–Ampère operator on quaternionic space

“…It has already been solved on compact manifolds with a flat hyperKähler metric [5]. Relating to this problem, it is meaningful and interesting to study the quaternionic Monge-Ampère operator and develop pluripotential theory on quaternionic manifolds [1,4,6,16,22]. The quaternionic Monge-Ampère operator is defined as the Moore determinant of the quaternionic Hessian of u:…”

Mixed Hessian inequalities and uniqueness in the class $$\mathcal {E}(X,\omega ,m)$$ E ( X , ω , m )

“…The complex Hessian equation on compact Kähler manifolds is not that geometric because the solutions do not yield Kähler metrics (see hovewer [1] for some geometric applications). Nevertheless the PDE theory is interesting on its own right, and a strong motivation for considering it is its real counterpart that has been developed some time ago thanks to the works of Trudinger, Wang, Chou and others (see [8,11,22,23,26] and references therein).…”

Balanced HKT metrics and strong HKT metrics on hypercomplex manifolds

“…It is easy to see that in this case, Ω + ∂∂ J ϕ is an HKT-form (see the proof of Proposition 4.2), hence Conjecture 4.1 implies existence of an HKT-metric Ω ′ = Ω + ∂∂ J ϕ such that the corresponding volume form is proportional to Φ. When Hol(M ) ⊂ SL(n, H), this conjecture was partly verified in [AV2]. We have shown that the solution of (4.1) is unique, and also gave a priori C 0 -bounds on its solution.…”

“…The notion of positive (2p, 0)-forms on hypercomplex manifolds (sometimes called q-positive, or H-positive) was developed in [V4] and [AV1] (see also [AV2] and [V8]). For our present purposes, only (2, 0)-forms are interesting, but everything can be immediately generalized to a general situation Let η ∈ Λ p,q I (M ) be a differential form.…”

“…The SL(n, H)-manifolds were studied in [AV2] and [V8], because on such manifolds the quaternionic Dolbeault complex is identified with a part of de Rham complex (Proposition 3.1). Under this identification, H-positive forms become positive in the usual sense, and ∂, ∂ J -closed or exact forms become ∂, ∂-closed or exact.…”