A note on equivariant biharmonic maps and stable biharmonic maps

TL;DR

This paper extends the theory of biharmonic maps to equivariant maps between model spaces, classifies certain conformal biharmonic maps, and proves the nonexistence of stable proper biharmonic maps with specific properties.

Contribution

It generalizes biharmonic equations for equivariant maps, classifies conformal biharmonic maps in 4D, and establishes nonexistence results for stable proper biharmonic maps.

Findings

Complete classification of conformal biharmonic maps from 4D space forms.

Generalization of biharmonic equations for equivariant maps.

Proof of nonexistence of stable proper biharmonic maps with constant tension norm.

Abstract

In this note, we generalize biharmonic equation for rotationally symmetric maps ([4], [16], [10]) to equivariant maps between model spaces and use it to give a complete classification of rotationally symmetric conformal biharmonic maps from a $4$-dimensional space form into a $4$-dimensional model space. We also give an improved second variation formula for biharmonic maps into a space form and use it to prove that there exists no stable proper biharmonic maps with constant square norm of tension field from a compact Riemannian manifold without boundary into a space form of positive sectional curvature.