Remarks on constructing biharmonic and conformal biharmonic maps to spheres

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.

💡 Research Summary

The paper investigates two fourth‑order generalisations of harmonic maps—biharmonic and conformal‑biharmonic maps—when the target is a Euclidean sphere. Starting from a harmonic map (v) into a lower‑dimensional sphere, the author introduces a simple geometric ansatz that “tilts’’ the map toward the equator of a higher‑dimensional sphere: \