A family of triharmonic maps to spheres in all dimensions greater than two
Volker Branding, Anna Siffert
TL;DR
The paper advances explicit constructions of higher-order harmonic maps into spheres by combining Nakauchi's higher-order radial maps with rotations of eigenmaps. It proves existence results for countably many proper biharmonic and triharmonic maps from $\mathbb{R}^m\setminus\{0\}$ to spheres for all $m\ge 3$, via solvability conditions on angular parameters and, in the triharmonic case, polynomial constraints. It also develops a general deformation framework to generate proper $r$-harmonic maps on spheres from eigenmaps, yielding unstable yet explicit families. Together, these contributions expand the catalog of explicit solutions to sixth-order PDEs in differential geometry and enhance understanding of polyharmonic variational problems on spheres.
Abstract
We present a construction method for triharmonic maps to spheres. In particular, we show that for any $m\in\mathbb{N}$ with $m\geq 3$ there exists a triharmonic map from $\mathbb{R}^m\setminus\{0\}$ into a round sphere. In addition, we provide a construction method for proper $r$-harmonic maps between spheres based on a suitable deformation of eigenmaps.