A characterization of the Legendre involution on the class of generic frontals

TL;DR

The paper characterizes the Legendre involution among generic frontals, proving its uniqueness under certain conditions and exploring its complexification.

Contribution

It establishes that any involution with the same fixed points as the Legendre involution must be the Legendre involution itself, under mild assumptions.

Findings

Uniqueness of the Legendre involution among generic frontals

Identification of the fixed point set of the Legendre involution

Complexification of the Legendre involution

Abstract

We show that, under an additional mild assumption, on the class of generic frontals, any involution whose fixed point set is exactly the same as the fixed point set of the Legendre involution must be the Legendre involution (Theorem 2 in \S 1). Moreover, its natural complexification (Corollary 1 in \S 3) is simultaneously shown.