Octonionic Calabi-Yau theorem

TL;DR

This paper introduces octonionic Kähler metrics on 16-dimensional manifolds, formulates an octonionic Monge-Ampère equation, and proves an analogue of the Calabi-Yau theorem in this new geometric setting.

Contribution

It defines a new class of octonionic Kähler metrics and establishes an existence theorem for the associated Monge-Ampère equation, extending Calabi-Yau theory to octonionic geometry.

Findings

Introduction of octonionic Kähler metrics

Formulation of octonionic Monge-Ampère equation

Proof of Calabi-Yau type theorem in octonionic setting

Abstract

A new class of Riemannian metrics, called octonionic K\"ahler, is introduced and studied on a certain class of 16-dimensional manifolds. It is an octonionic analogue of K\"ahler metrics on complex manifolds and of HKT-metrics of hypercomplex manifolds. Then for this class of metrics an octonionic version of the Monge-Amp\`ere equation is introduced and solved under appropriate assumptions. The latter result is an octonionic version of the Calabi-Yau theorem from K\"ahler geometry.