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On the envelope of Poisson functional on almost complex manifolds

Florian Bertrand, Uroš Kuzman

TL;DR

This work extends Poletsky’s disc-analytic framework to smooth almost complex manifolds of arbitrary complex dimension by developing a robust gluing method for $J$-holomorphic discs. Central to the method is a local Newton-type iteration with a uniformly bounded right inverse for the linearized operator, together with a nonlinear Cousin problem that enables gluing across multiple charts, and a Riemann–Hilbert-type attachment to tori. The main result establishes that the envelope $EP_f$ of the Poisson functional is either identically $- ablafty$ or $J$-plurisubharmonic, generalizing prior complex-analytic results to higher-dimensional almost complex settings. These techniques yield three applications: regularization of $J$-plurisubharmonic functions, a disc-based characterization of $\mathrm{Psh}_J$-hulls, and an envelope formula for the Lelong functional, broadening the scope of disc methods in almost complex geometry and potential theory.

Abstract

We establish the plurisubharmonicity of the envelope of the Poisson functional on almost complex manifolds. That is, we generalize the corresponding result for complex manifolds and almost complex manifolds of complex dimension two.

On the envelope of Poisson functional on almost complex manifolds

TL;DR

This work extends Poletsky’s disc-analytic framework to smooth almost complex manifolds of arbitrary complex dimension by developing a robust gluing method for -holomorphic discs. Central to the method is a local Newton-type iteration with a uniformly bounded right inverse for the linearized operator, together with a nonlinear Cousin problem that enables gluing across multiple charts, and a Riemann–Hilbert-type attachment to tori. The main result establishes that the envelope of the Poisson functional is either identically or -plurisubharmonic, generalizing prior complex-analytic results to higher-dimensional almost complex settings. These techniques yield three applications: regularization of -plurisubharmonic functions, a disc-based characterization of -hulls, and an envelope formula for the Lelong functional, broadening the scope of disc methods in almost complex geometry and potential theory.

Abstract

We establish the plurisubharmonicity of the envelope of the Poisson functional on almost complex manifolds. That is, we generalize the corresponding result for complex manifolds and almost complex manifolds of complex dimension two.
Paper Structure (9 sections, 9 theorems, 59 equations)

This paper contains 9 sections, 9 theorems, 59 equations.

Table of Contents

  1. Gluing techniques for $J$-holomorphic maps
  2. The local Newton-type iteration
  3. The non-linear Cousin problem
  4. Approximately attaching a disc to a torus
  5. Proof of Theorem 1
  6. Applications
  7. Regularization of $J$-plurisubharmonic functions
  8. $\mathrm{Psh}_J$-hull and Poletsky discs
  9. Envelope of the Lelong functional

Key Result

Theorem 1

Let $f\colon M\to\mathbb{R}\cup\left\{-\infty\right\}$ be an upper semi-continuous function defined on a smooth almost complex manifold $(M,J)$. Then $EP_f$ is $J$-plurisubharmonic on $M$ or identically $-\infty$.

Theorems & Definitions (13)

  • Theorem 1
  • Definition 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • proof
  • Lemma 6
  • proof : Proof of Lemma \ref{['c']}
  • proof : Proof of Theorem \ref{['main theorem']}
  • Corollary 7
  • ...and 3 more