A Uniform Estimate for the Quaternionic Gauduchon Metric With Prescribed Volume Form

Abstract The quaternionic Calabi conjecture, posed by Alesker and Verbitsky [1], predicts that the quaternionic Monge–Ampère equation can always be solved on any compact HKT manifold. Motivated by this conjecture, we will introduce a quaternionic version of the Gauduchon conjecture on any compact $SL(n,\mathbb{H})$-manifold, specifically addressing the existence of quaternionic Gauduchon metrics with prescribed volume form. We reframe this question as a special case of fully nonlinear elliptic equations of second order and subsequently establish a uniform estimate for the potential function.