Pluri-Potential Theory, Submersions and Calibrations

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 Kähler and \(G_2\) $G_2$ manifolds, emphasizing both parallels and differences. We show that previous results concerning Lagrangian fibrations can be viewed as an application of this framework.