Rotationally symmetric biharmonic maps between models

TL;DR

This paper investigates the existence, classification, and stability of rotationally symmetric proper biharmonic maps between models, focusing on conformal diffeomorphisms in four dimensions and their stability under equivariant variations.

Contribution

It provides a complete classification of rotationally symmetric proper biharmonic conformal diffeomorphisms in four dimensions and analyzes their stability properties.

Findings

Complete classification of biharmonic conformal diffeomorphisms in 4D models.

Identification of stable inverse stereographic projections.

Analysis of stability with respect to boundary-preserving variations.

Abstract

The main aim of this paper is to study existence and stability properties of rotationally symmetric proper biharmonic maps between two $m$-dimensional models (in the sense of Greene and Wu). We obtain a complete classification of rotationally symmetric, proper biharmonic conformal diffeomorphisms in the special case that $m=4$ and the models have constant sectional curvature. Then, by introducing the Hamiltonian associated to this problem, we also obtain a complete description of conformal proper biharmonic solutions in the case that the domain model is ${\mathbb R}^4$. In the second part of the paper we carry out a stability study with respect to equivariant variations (equivariant stability). In particular, we prove that: (i) the inverse of the stereographic projection from the open $4$-dimensional Euclidean ball to the hyperbolic space is equivariant stable; (ii) the inverse of the stereographic projection from the closed $4$-dimensional Euclidean ball to the sphere is equivariant stable with respect to variations which preserve the boundary data.