New examples of CR submanifolds for the $2$-jet determination problem

TL;DR

This paper explores the 2-jet determination problem for CR automorphisms, providing new examples of submanifolds where automorphisms are uniquely determined by their 2-jets, advancing understanding in CR geometry.

Contribution

It introduces a novel example of a non-strictly pseudoconvex CR submanifold with 2-jet determination of automorphisms, expanding known classes of such manifolds.

Findings

New example of a CR submanifold with 2-jet determination

Analysis of 2-jet determination in pseudoconvex hypersurfaces

Formulation of a general problem with illustrative examples

Abstract

This paper addresses two questions related to mapping problems. In the first part of the paper, we discuss some recent results regarding the $2$-jet determination for biholomorphisms between smooth (weakly) pseudoconvex hypersurfaces; in light of those, we formulate a general problem and illustrate it with examples. In the second part of the paper, we provide a new example of a real submanifold of codimension $7$ in $\mathbb{C}^{10}$ that is not strictly pseudoconvex and for which its germs of CR automorphisms are determined by their $2$-jets at a given point.