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Analytic structure of $q$-pseudoconcave subsets of continuous graphs

Filippo Valnegri

Abstract

Let $Γ(g)$ be the graph of a continuous function $g:D\subset\mathbb{C}^n\times\mathbb{R}\to\mathbb{R}\times\mathbb{C}^p$ (for $n\geq 1$ and $p\geq 0$). We prove that any $n$-pseudoconcave subset $Z$ of $Γ(g)$ can be realised as the disjoint union of $n$-dimensional complex manifolds. Moreover, we show that the same conclusion can be made for any $n$-pseudoconcave subset $Z$ of $\mathbb{C}^N$ (for $1\leq n<N$), which is locally the graph of a continuous function over a closed subset of $\mathbb{C}^n\times\mathbb{R}$.

Analytic structure of $q$-pseudoconcave subsets of continuous graphs

Abstract

Let be the graph of a continuous function (for and ). We prove that any -pseudoconcave subset of can be realised as the disjoint union of -dimensional complex manifolds. Moreover, we show that the same conclusion can be made for any -pseudoconcave subset of (for ), which is locally the graph of a continuous function over a closed subset of .
Paper Structure (9 sections, 29 theorems, 163 equations)

This paper contains 9 sections, 29 theorems, 163 equations.

Table of Contents

  1. Acknowledgments.
  2. The hypersurface case
  3. Some properties of q-local maximum sets
  4. Upper closed limit and local maximum property
  5. The higher codimension case
  6. Case $\{\alpha_0\}=I$.
  7. Case $\alpha_0\in\mathring{I}$.
  8. Case $\alpha_0\in bI$.
  9. Some applications

Key Result

Theorem 1

Let $D\subset\mathbb{C}^n\times\mathbb{R}$ be a domain and let $g:D\to\mathbb{R}$ be a $\mathscr{C}^2$ function. If $z_0\in\Gamma(g)$ is not in the envelope of holomorphy of neither $\Gamma_+(g)$ nor $\Gamma_-(g)$, then there exists a complex hypersurface which passes through $z_0$ and is contained

Theorems & Definitions (68)

  • Theorem 1: Trépreau trepreau:prolongement
  • Theorem 2: Shcherbina shcherbina:polynomial_hull
  • Theorem 3: Chirka chirka:trepreau_theorem
  • Theorem 3$^{\prime}$
  • Theorem 4: Pawlaschyk, Shcherbina pawl_shcher:foliations
  • Remark 1
  • Remark 2
  • Theorem 4$^{\prime}$: pawl_shcher:foliations
  • Theorem 5: Słodkowski slodkowski:loc_max_property
  • Corollary 1
  • ...and 58 more