A general approach to equivariant biharmonic maps

TL;DR

This paper introduces a 1-dimensional variational method for constructing equivariant biharmonic maps, providing analytical conditions, stability results, and insights into their properties compared to harmonic maps.

Contribution

It presents a novel direct variational approach for equivariant biharmonic maps and explores their stability and maximum principle properties.

Findings

Derived analytical conditions for biharmonicity with symmetries

Established a 1-dimensional stability result

Showed biharmonic maps do not satisfy the classical maximum principle

Abstract

In this paper we describe a 1-dimensional variational approach to the analytical construction of equivariant biharmonic maps. Our goal is to provide a direct method which enables analysts to compute directly the analytical conditions which guarantee biharmonicity in the presence of suitable symmetries. In the second part of our work, we illustrate and discuss some examples. In particular, we obtain a 1-dimensional stability result, and also show that biharmonic maps do not satisfy the classical maximum principle proved by Sampson for harmonic maps.