[Paper Review] Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds

Semyon Alesker, Misha Verbitsky|arXiv (Cornell University)|Feb 28, 2008

Geometry and complex manifolds17 references4 citations

TL;DR

This paper introduces a quaternionic Monge-Ampère equation on compact hypercomplex manifolds equipped with an HKT-metric, posing a quaternionic analogue of the Calabi problem. It establishes uniqueness up to a constant and a zeroth-order a priori estimate for solutions under the condition that the holonomy of the Obata connection lies in $SL(n,\mathbb{H})$, with existence conjectured analogously to the Calabi-Yau theorem.

ABSTRACT

A quaternionic version of the Calabi problem on Monge-Ampere equation is introduced. It is a quaternionic Monge-Ampere 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 solution is conjectured, similar to Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of a complex Hessian equation, making sense on any complex manifold.

Motivation & Objective

• To formulate a quaternionic version of the Calabi problem for HKT-manifolds, analogous to the complex Calabi-Yau theorem.
• To study the solvability of a non-linear elliptic second-order equation involving the quaternionic Monge-Ampère operator on compact hypercomplex manifolds.
• To establish uniqueness (up to a constant) and a zeroth-order a priori estimate for solutions under the holonomy condition $\mathrm{Hol}(\nabla_{\mathrm{Obata}}) \subset SL(n,\mathbb{H})$.
• To reformulate the quaternionic equation as a special case of the complex Hessian equation, valid on any complex manifold.

Proposed method

• Define the quaternionic Monge-Ampère equation as $(\Omega + \partial\partial_J\varphi)^n = e^f \Omega^n$, where $\Omega$ is an HKT-form and $\varphi$ is a real smooth function.
• Use the operators $R$ and $V$ to relate the quaternionic structure to the complex Dolbeault calculus on the complex manifold $(M,I)$.
• Reformulate the quaternionic equation as a complex Hessian equation on the complex manifold $(M,I)$, enabling the use of complex geometric techniques.
• Prove a zeroth-order a priori estimate via a modified Yau-type argument, using $L^p$-norms and Sobolev inequalities on the manifold.
• Establish $L^p$-bounds on $\varphi$ and its gradient, leveraging the spectral gap of the Laplacian under the normalization $\int_M \varphi \cdot \Omega^n \wedge \overline{\Theta} = 0$.
• Apply Moser iteration and interpolation techniques to derive uniform $C^0$-bounds on $\varphi$ from $L^p$-bounds as $p \to \infty$.

Experimental results

Research questions

• RQ1 Does the quaternionic Monge-Ampère equation $(\Omega + \partial\partial_J\varphi)^n = e^f \Omega^n$ admit a smooth solution on a compact hypercomplex manifold with HKT-metric when the holonomy lies in $SL(n,\mathbb{H})$?
• RQ2 Under what conditions is the solution to the quaternionic Monge-Ampère equation unique up to a constant?
• RQ3 Can the zeroth-order a priori estimate $||\varphi||_{C^0} \leq C$ be established for solutions under normalization, given $f \in C^0(M)$?
• RQ4 How can the quaternionic Monge-Ampère equation be reformulated as a complex Hessian equation on a complex manifold?

Key findings

• Uniqueness of the solution to the quaternionic Monge-Ampère equation is proven up to a constant under the condition $\mathrm{Hol}(\nabla_{\mathrm{Obata}}) \subset SL(n,\mathbb{H})$.
• A zeroth-order a priori estimate $||\varphi||_{C^0} \leq C$ holds for solutions normalized by $\int_M \varphi \cdot \Omega^n \wedge \overline{\Theta} = 0$, where $C$ depends only on $M$, $\Omega$, and $||f||_{C^0}$.
• The solution satisfies $||\varphi||_{L^p} \leq Q_1 (C_3 p)^{-2n/p}$ for all $p \geq 2$, with constants depending only on $M$, $g_0$, and $||f||_{C^0}$.
• The $L^p$-bounds on $\varphi$ and $\nabla\varphi$ are established via Sobolev and Poincaré-type inequalities, leveraging the spectral gap of the Laplacian.
• The $C^0$-bound is derived as the limit $||\varphi||_{C^0} = \lim_{p \to \infty} ||\varphi||_{L^p} \leq Q_1$, confirming uniform boundedness.
• The quaternionic Monge-Ampère equation is reformulated as a complex Hessian equation on the complex manifold $(M,I)$, extending its applicability to general complex manifolds.