Rotationally symmetric biharmonic maps between models

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.