Some recent work on biharmonic conformal maps

TL;DR

This paper reviews recent developments in biharmonic conformal maps, focusing on their properties, links to isoparametric functions, and connections to Yamabe equations, including various subclasses like morphisms and submersions.

Contribution

It provides a comprehensive survey of recent advances in biharmonic conformal maps and explores their relationships with other geometric and analytical concepts.

Findings

Connections between biharmonic conformal maps and isoparametric functions.

Relations to Yamabe type equations.

Overview of biharmonic morphisms and submersions.

Abstract

This note reviews some of the recent work on biharmonic conformal maps (see \cite{OC}, Chapter 11, for a detailed survey). It will be focused on biharmonic conformal immersions and biharmonic conformal maps between manifolds of the same dimension and their links to isoparametric functions and Yamabe type equations, though biharmonic morphisms (maps that preserve solutions of bi-Laplace equations), generalized harmonic morphisms (maps that pull back germs of harmonic functions to germs of biharmonic functions), and biharmonic conformal and Riemannian submersions will also be touched.