On conformal points of area preserving maps and related topics

TL;DR

This paper investigates the existence of conformal points in area-preserving maps, linking the problem to classical conjectures and providing conditions for their existence in specific classes of symplectomorphisms.

Contribution

It introduces new conditions guaranteeing conformal points for Hamiltonian vector fields and moderate symplectomorphisms, connecting the problem to longstanding conjectures.

Findings

Conditions for existence of conformal points in Hamiltonian vector fields

Conditions for conformal points in moderate symplectomorphisms

Connection to Carathéodory and Loewner conjectures

Abstract

In this article we consider area preserving diffeomorphisms of planar domains, and we are interested in their conformal points, i.e., points at which the derivative is a similarity. We present some conditions that guarantee existence of conformal points for the infinitesimal problem of Hamiltonian vector fields as well as for what we call moderate symplectomorphisms of simply connected domains. We also link this problem to the Carath\'eodory and Loewner conjectures.