Stationary holomorphic discs and finite jet determination problems

TL;DR

This paper develops a biholomorphically invariant method using stationary holomorphic discs to address finite jet determination problems for Levi non-degenerate hypersurfaces, even with limited smoothness.

Contribution

It introduces a new technique involving stationary holomorphic discs that enables finite jet determination results under weaker regularity assumptions.

Findings

Achieved 2-jet determination for germs of biholomorphisms and CR diffeomorphisms.

Extended finite jet determination results to the almost complex setting.

Provided a globally biholomorphically invariant construction of analytic discs.

Abstract

We construct a family of small analytic discs attached to Levi non-degenerate hypersurfaces in $\mathbb{C}^{n+1}$, which is globally biholomorphically invariant. We then apply this technique to study unique determination problems along Levi non-degenerate hypersurfaces that are merely of class $\mathcal{C}^4$. This method gives 2-jet determination results for germs of biholomorphisms, CR diffeomorphisms, as well as in the almost complex setting.