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Generalized stationary discs attached to degenerate submanifolds in $\mathbb{C}^N$

Al Masri Mohammad Tarek, Bertrand Florian, Meylan Francine, Oueidat Lea, Zoghaib Hadi

Abstract

We study the family of generalized stationary discs attached to a Levi degenerate submanifold M of codimension d in $\mathbb{C}^{n+d}$. We show, under suitable geometric assumptions on M, that this family forms a finite dimensional real submanifold of the Banach space of analytic discs.

Generalized stationary discs attached to degenerate submanifolds in $\mathbb{C}^N$

Abstract

We study the family of generalized stationary discs attached to a Levi degenerate submanifold M of codimension d in . We show, under suitable geometric assumptions on M, that this family forms a finite dimensional real submanifold of the Banach space of analytic discs.
Paper Structure (8 sections, 4 theorems, 78 equations)

This paper contains 8 sections, 4 theorems, 78 equations.

Table of Contents

  1. Admissible degenerate submanifolds of $\mathbb C^{n+d}$
  2. Generalized stationary discs and the Riemann-Hilbert Problem
  3. Function spaces
  4. Generalized stationary discs
  5. A Riemann-Hilbert type problem
  6. Construction of generalized stationary discs and its consequences
  7. The main result
  8. Finite jet determination of generalized stationary lifts and open questions

Key Result

Theorem 3.1

Let $M_H=\{\rho=0\} \subset \mathbb C^{n+d}$ be a model submanifold of the form (eqmod). Consider an initial stationary lift $\bm{f_0}=\left(h_0,g_0,\tilde{h_0},\tilde{g_0}\right) \in \mathcal{S}^{k_0}_0(M_H)$ of the form eqdisini where $(c_1,\ldots,c_d) \in \mathbb R^d$ and $V\in \mathbb C^n\setmin such that: In particular, for any defining function $r \in U$, the set is a $\mathcal{C}^1$ real

Theorems & Definitions (17)

  • Definition 1.1
  • Remark 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Definition 2.1
  • Example 2.2
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['theodiscs']}
  • ...and 7 more