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Foliations of continuous q-pseudoconcave graphs

Thomas Pawlaschyk, Nikolay Shcherbina

TL;DR

The paper addresses when the graph of a continuous map $f:D \to \mathbb{R}^k\times\mathbb{C}^p$ is locally foliated by complex $n$-dimensional submanifolds, linking this to Rothstein $n$-pseudoconvexity of the complement. It develops and applies the theory of $q$-plurisubharmonic functions and $q$-pseudoconvex/concave sets, together with a duality principle, to prove a general foliation theorem for graphs of continuous maps in the regime $k\in\{0,1\}$. The main result shows that if the graph is $n$-pseudo-concave, then locally it splits into $n$-dimensional complex leaves, with a detailed case analysis reducing to classical Hartogs, Sh93, and Chirka-type arguments or holomorphy of components along leaves. The work extends classical Hartogs and Levi-type phenomena to higher codimension graphs, clarifying how $q$-pseudoconvexity of the complement governs complex foliations and providing a robust framework for future geometric-analytic investigations in several complex variables.

Abstract

We show that for $k = 0, 1$ the graph of a continuous mapping $f:D \to \mathbb{R}^k\times\mathbb{C}^p$, defined on a domain $D$ in $\mathbb{C}^n\times\mathbb{R}^k$, is locally foliated by complex $n$-dimensional submanifolds if and only if its complement is $n$-pseudoconvex (in the sense of Rothstein) relatively to $(D\times\mathbb{R}^k)\times\mathbb{C}^p\subset \mathbb{C}^{n}\times\mathbb{C}^k\times\mathbb{C}^p$.

Foliations of continuous q-pseudoconcave graphs

TL;DR

The paper addresses when the graph of a continuous map is locally foliated by complex -dimensional submanifolds, linking this to Rothstein -pseudoconvexity of the complement. It develops and applies the theory of -plurisubharmonic functions and -pseudoconvex/concave sets, together with a duality principle, to prove a general foliation theorem for graphs of continuous maps in the regime . The main result shows that if the graph is -pseudo-concave, then locally it splits into -dimensional complex leaves, with a detailed case analysis reducing to classical Hartogs, Sh93, and Chirka-type arguments or holomorphy of components along leaves. The work extends classical Hartogs and Levi-type phenomena to higher codimension graphs, clarifying how -pseudoconvexity of the complement governs complex foliations and providing a robust framework for future geometric-analytic investigations in several complex variables.

Abstract

We show that for the graph of a continuous mapping , defined on a domain in , is locally foliated by complex -dimensional submanifolds if and only if its complement is -pseudoconvex (in the sense of Rothstein) relatively to .

Paper Structure

This paper contains 13 sections, 16 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. On $q$-Plurisubharmonic Functions
  3. On $q$-Pseudoconvex Sets
  4. Duality Principle of $q$-Pseudoconvex Sets
  5. On $q$-Pseudoconcave Graphs
  6. Case $\mathbf{n \geq 1}$, $\mathbf{k=0}$, $\mathbf{p=1}$.
  7. Case $\mathbf{n \geq 1}$, $\mathbf{k=0}$, $\mathbf{p\geq 1}$.
  8. Case $\mathbf{n = 1}$, $\mathbf{k=1}$, $\mathbf{p=0}$.
  9. Case $\mathbf{n \geq 1}$, $\mathbf{k=1}$, $\mathbf{p=0}$.
  10. Case $\mathbf{n = 1}$, $\mathbf{k=1}$, $\mathbf{p = 1}$.
  11. Case $\mathbf{n \geq 1 }$, $\mathbf{k=1}$, $\mathbf{p=1}$.
  12. Case $\mathbf{n = 1}$, $\mathbf{k=1}$, $\mathbf{p \geq 1}$.
  13. Case $\mathbf{n \geq 1}$, $\mathbf{k=1}$, $\mathbf{p \geq 1}$.

Key Result

Theorem 1.1

Let $f:B\to\mathbb{C}_\zeta$ be a continuous function defined on the unit ball $B \subset\mathbb{C}^n_z$. Then the complement of its graph $\Gamma(f)=\{(z,\zeta): z \in B, \zeta=f(z)\}$ in $B\times\mathbb{C}_\zeta$ is a domain of holomorphy if and only if the function $f$ is holomorphic.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.5: Local maximum property
  • Definition 2.6
  • Theorem 2.7: Local maximum principle for analytic sets
  • ...and 26 more