Reduction methods for the bienergy

TL;DR

This paper develops reduction methods based on symmetries to find critical points of the bienergy, simplifying complex PDEs to ODEs, and explores their application to biharmonic conformal maps and G-invariant immersions.

Contribution

It introduces reduction techniques for the bienergy using symmetries, enabling the construction of new examples of biharmonic maps and immersions.

Findings

Reduction methods simplify the analysis of biharmonic maps.

Examples of G-invariant immersions are constructed.

Open problems and future research directions are discussed.

Abstract

This paper, in which we develop ideas introduced in \cite{MR}, focuses on \emph{reduction methods} (basically, group actions or, more generally, simmetries) for the bienergy. This type of techniques enable us to produce examples of critical points of the bienergy by reducing the study of the relevant fourth order PDE's system to ODE's. In particular, we shall study rotationally symmetric biharmonic conformal diffeomorphisms between \emph{models}. Next, we will adapt the reduction method to study an ample class of $G-$invariant immersions into the Euclidean space. At present, the known instances in these contexts are far from reaching the depth and variety of their companions which have provided fundamental solutions to classical problems in the theories of harmonic maps and minimal immersions. However, we think that these examples represent an important starting point which can inspire further research on biharmonicity. In this order of ideas, we end this paper with a discussion of some open problems and possible directions for further developments.