Pluri-potential theory, submersions and calibrations

TL;DR

This paper explores the interactions of convex, subharmonic, and pluri-subharmonic functions on manifolds connected by Riemannian submersions, extending Harvey-Lawson's pluri-potential theory to calibrated manifolds like Kähler and G2 types.

Contribution

It systematically extends classical complex-analytic results to the setting of calibrated manifolds, highlighting parallels and differences, and applies the framework to Lagrangian fibrations.

Findings

Extended pluri-potential theory to Kähler and G2 manifolds

Established parallels between complex and calibrated geometries

Applied results to Lagrangian fibrations

Abstract

We present a systematic collection of results concerning interactions between convex, subharmonic and pluri-subharmonic functions on pairs of manifolds related by a Riemannian submersion. Our results are modelled on those known in the classical complex-analytic context and represent another step in the recent Harvey-Lawson pluri-potential theory for calibrated manifolds. In particular we study the case of K\"ahler and G2 manifolds, emphasizing both parallels and differences. We show that previous results concerning Lagrangian fibrations can be viewed as an application of this framework.