Biharmonic conformal maps in dimension four and equations of Yamabe-type

TL;DR

This paper links biharmonic conformal maps in four dimensions to a Yamabe-type equation, enabling the construction and characterization of solutions on specific Einstein manifolds, including the Euclidean 4-sphere.

Contribution

It reduces the biharmonic conformal map problem to a Yamabe-type equation and constructs explicit examples on the Euclidean 4-sphere, also characterizing solutions on Euclidean 4-space.

Findings

Infinite family of solutions on Euclidean 4-sphere

Characterization of solutions on Euclidean 4-space

Existence of non-constant proper biharmonic conformal maps from Einstein 4-manifolds

Abstract

We prove that the problem of constructing biharmonic conformal maps on a $4$-dimensional Einstein manifold reduces to a Yamabe-type equation. This allows us to construct an infinite family of examples on the Euclidean 4-sphere. In addition, we characterize all solutions on Euclidean 4-space and show that there exists at least one non-constant proper biharmonic conformal map from any closed Einstein 4-manifold of negative Ricci curvature.