A hypercomplex manifold [C. P. Boyer, Proc. Am. Math. Soc. 102, No. 1, 157–164 (1988; Zbl 0642.53073)] is a smooth manifold \(M\) together with a triple \((I,J,K)\) of complex structures satisfying the usual quaternionic relations: \(IJ=-JI=K\). Necessarily, the real dimension of a hypercomplex manifold is divisible by 4. In the paper it is presumed that the complex structures \(I,J,K\) act on the right on the tangent bundle \(TM\) of \(M\). This action extends uniquely to the right action of the algebra \(\mathbb{H}\) of quaternions on \(TM\). Let \(M\) be a hypercomplex manifold, and \(G\) a Riemannian metric on \(M\). The metric \(g\) is called quaternionic Hermitian (or hyper-Hermitian) if \(g\) is invariant with respect to the group \(SU(2)\subset \mathbb{H}^\ast\) of unitary quaternions.
Given a quaternionic Hermitian metric \(g\) on a hypercomplex manifold \(M\), the differential form \(\Omega:=-\omega_J+\sqrt{-1}\omega_K\) is considered, where \(\omega_L(A,B):=g(A,B\circ L)\) for any \(L\in \mathbb{H}\) with \(L^2=-1\) and any vector fields \(A,B\) on \(M\). Then \(\Omega\) is a \((2,0)\)-form with respect to the complex structure \(I\). The metric \(g\) on \(M\) is called an HKT-metric (Hyper Kähler with Torsion) if \(\partial \Omega=0\), where \(\partial\) is the usual \(\partial\)-differential on the complex manifold \((M,I)\). HKT-metrics on hypercomplex manifolds first were introduced by P. S. Howe and G. Papadopoulos [Phys. Lett. B 379, 80–86 (1996)].
HKT-metrics on hypercomplex manifolds are analogous in many respects to Kähler metrics on complex manifolds. For example, if any Kähler form \(\omega\) on a complex manifold can be locally written in the form \(\omega=dd^ch\) where \(h\) is strictly plurisubharmonic function called a potential of \(\omega\), and vice versa, an HKT-form \(\Omega\) on a hypercomplex manifold locally admits a potential: it can be written as \(\omega=\partial\partial_JH\), where \(\partial_J=J^{-1}\circ\overline{\partial}\circ J\), and \(H\) is strictly plurisubharmonic function in the quaternionic sense; and the converse is also true.
Motivated by the analogy with the complex case, the authors of the present paper introduce the following quaternionic version of the Calabi problem. Let \((M^{4n},I,J,K)\) be a compact hypercomplex manifold of real dimension \(4n\). Let \(\Omega\) be an HKT-form. Let \(f\) be a real-valued \(C^\infty\) function on \(M\). The quaternionic Calabi problem is to study solvability of the following quaternionic Monge-Ampère equation with an unknown real-valued function \(\varphi\):
\[ (\Omega+\partial\partial_J\varphi)^n=e^f\Omega^n. \]
The authors prove the following lemma.
Lemma 4.9. Assume that \(\varphi\) satisfies the quaternionic Monge-Ampère equation \((\Omega+\partial\partial_J\varphi)^n=e^f\Omega^n\) on a compact manifold \(M\). Then the form \(\Omega+\partial\partial_J\varphi\) belongs to the interior of the cone of (strongly=weakly) \(q\)-positive (2,0)-forms. Hence \(\Omega+\partial\partial_J\varphi\) is an HKT-form.
This means that \(\Omega+\partial\partial_J\varphi\) corresponds to a new HKT-metric. This equation is non-linear elliptic equation of second order. The authors formulate the following conjecture.
Conjecture 1.5. Let us assume that \((M,I)\) admits a holomorphic (with respect to the complex structure \(I\)) non-vanishing \((2n,0)\)-form \(\Theta\). Then the quaternionic Monge-Ampère equation has a \(C^\infty\)-solution \(\varphi\) provided the following necessary condition on the initial data is satisfied:
\[ \int_M(e^f-1)\Omega^n\wedge \overline{\Theta}=0. \]
Then the authors show that under the condition of existence of such \(\Theta\) a solution of the quaternionic Monge-Ampère equation is unique up to a constant.
The main result of the paper is the following:
Corollary 5.7. There exists a constant \(C\) depending on \(M,\Omega\), and \(\|f\|_{C^0}\) only, such that the solution \(\varphi\) satisfying the normalization condition \(\int_M \varphi.\Omega^n\wedge \overline{\Theta}=0\) must satisfy the estimate \(\|\varphi\|_{C^0}\leq C\), where \(\|.\|_{C^0}\) is the maximum norm on \(M\), i.e., \(\|u\|_{C^0}:=max\{|u(x)||x\in M\}\).
The proof of this result is a modification of S.-T. Yau’s argument [Commun. Pure Appl. Math. 31, 339–411 (1978; Zbl 0369.53059)] in the complex case as presented by D. D. Joyce in [Compact manifolds with special holonomy. Oxford: Oxford University Press (2000; Zbl 1027.53052)].