The 1-jet determination of stationary discs attached to generic CR submanifolds

TL;DR

This paper provides an elementary proof for the equivalence of stationary disc existence and 1-jet determination in strongly pseudoconvex CR submanifolds, extending previous results and proposing a conjecture for more general cases.

Contribution

It offers a new elementary proof of a key equivalence in CR geometry and suggests a broader conjecture beyond strongly pseudoconvex models.

Findings

Elementary proof of the equivalence when the model is strongly pseudoconvex

Explicit expression of stationary discs used in the proof

Conjecture proposed for non-strongly pseudoconvex models

Abstract

The existence of a nondefective stationary disc attached to a nondegenerate model quadric in C^N is a necessary condition to ensure the unique 1-jet determination of the lifts of a key family of stationary discs. In this paper, we give an elementary proof of the equivalence when the model quadric is strongly pseudoconvex, recovering a result of Tumanov. Our proof is based on the explicit expression of stationary discs, and opens up a conjecture for the unique 1-jet determination to hold when the model is not necessarily strongly pseudoconvex.