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On Horizontal Immersions of Discs in Fat Distributions of Type $(4,6)$

Aritra Bhowmick

Abstract

In this article we discuss horizontal immersions of discs in certain corank-$2$ fat distributions on $6$-dimensional manifolds. The underlying real distribution of a holomorphic contact distribution on a complex $3$ manifold belongs to this class. The main result presented here says that the associated nonlinear PDE is locally invertible. Using this we prove the existence of germs of embedded horizontal discs.

On Horizontal Immersions of Discs in Fat Distributions of Type $(4,6)$

Abstract

In this article we discuss horizontal immersions of discs in certain corank- fat distributions on -dimensional manifolds. The underlying real distribution of a holomorphic contact distribution on a complex manifold belongs to this class. The main result presented here says that the associated nonlinear PDE is locally invertible. Using this we prove the existence of germs of embedded horizontal discs.

Paper Structure

This paper contains 19 sections, 17 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Immersions In a Manifold with a Distribution
  3. Horizontal Immersions in a Manifold with a Fat Distribution
  4. Main Results
  5. Basic Notions
  6. Bracket-Generating Distributions
  7. Fat Distributions
  8. Corank $2$ Fat Distributions
  9. Horizontal immersions in a corank 2 fat distribution
  10. Inversion of $\frL_f$ at $\calD$-horizontal Immersions
  11. Local Inversion of $\frD$
  12. Existence of Horizontal Germs and the Local $h$-Principle
  13. Proof of \ref{['lemma:jetLifting']}
  14. Convention for Multi-Indices :
  15. Induction Statement $\calP(\alpha)$ :
  16. ...and 4 more sections

Key Result

Theorem \oldthetheorem

There exists an open subset $\frU\subset C^\infty(\bbD^2,M)$ such that for every $f:\bbD^2\to M$ in $\frU$, the linearization $\frL_f$ of the operator $\frD$ at $f$ admits a tame right inverse.

Theorems & Definitions (46)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Example \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Example \oldthetheorem
  • Theorem \oldthetheorem: raynerFatmontTour
  • ...and 36 more