$L^\infty$ estimate for the potential of quaternionic Gauduchon metric with prescribed volume form
Abstract
The quaternionic Calabi conjecture, posed by Alesker and Verbitsky \cite{AleskerVerbitsky (2010)}, predicts that the quaternionic MongeAmpère 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2023
 DOI:
 10.48550/arXiv.2310.12597
 arXiv:
 arXiv:2310.12597
 Bibcode:
 2023arXiv231012597Z
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs
 EPrint:
 23 pages