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Pluri-potential theory, submersions and calibrations

Tommaso Pacini

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 Kähler 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.

Pluri-potential theory, submersions and calibrations

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 Kähler 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.
Paper Structure (36 sections, 18 theorems, 57 equations)

This paper contains 36 sections, 18 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Immersions vs. submersions.
  3. Pluri-potential theories.
  4. Submanifold geometry.
  5. Geometric constructions of PSH functions.
  6. Kähler vs. $G_2$ theory.
  7. Final comments.
  8. The Hessian operator
  9. Immersions.
  10. Submersions: generalities.
  11. Submersions: functions.
  12. Function theory and submersions: Model cases
  13. Potential theory.
  14. Classical pluri-potential theory.
  15. Conclusions.
  16. ...and 21 more sections

Key Result

Proposition 2.2

Let $\Sigma\hookrightarrow (M,g)$ be a submanifold endowed with the restricted metric.

Theorems & Definitions (33)

  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Definition 3.3
  • ...and 23 more