Remarks on constructing biharmonic and conformal biharmonic maps to spheres

TL;DR

This paper explores methods to construct biharmonic and conformal biharmonic maps into spheres, revealing restrictions for biharmonic maps on closed domains and demonstrating the potential for generating new conformal biharmonic maps.

Contribution

It introduces a geometric algorithm for constructing biharmonic and conformal biharmonic maps into spheres, highlighting differences based on domain compactness and providing explicit examples.

Findings

Maximum principle restricts biharmonic maps on closed domains

More flexibility exists for non-compact domains

Algorithm can produce new conformal biharmonic maps

Abstract

Biharmonic and conformal-biharmonic maps are two fourth-order generalizations of the well-studied notion of harmonic maps in Riemannian geometry. In this article we consider maps into the Euclidean sphere and investigate a geometric algorithm that aims at rendering a given harmonic map either biharmonic or conformally biharmonic. For biharmonic maps we find that in the case of a closed domain the maximum principle imposes strong restrictions on our approach, whereas there is more flexibility when we have a non-compact domain and we highlight this difference by a number of examples. Concerning conformal-biharmonic maps we show that our algorithm produces explicit critical points for maps between spheres. Moreover, it turns out that we do not get strong restrictions as we obtain for biharmonic maps, such that our algorithm might produce additional conformal-biharmonic maps between spheres beyond the ones found in this article.