Torus equivariant harmonic maps between spheres (Q5957463)

In the original article on harmonic maps between Riemannian manifolds [Am. J. Math. 86, 109-160 (1964; Zbl 0122.40102)], \textit{J. Eells} and \textit{J. H. Sampson} proved that from a compact domain into a space of negative curvature, harmonic representatives, in fact minimisers, exist in each and every homotopy class. NEWLINENEWLINENEWLINEWhen the target has points of positive sectional curvature, the existence is more problematic, and even the case of maps between spheres is still incomplete. The first techniques used, harmonic joins [\textit{R. T. Smith}, Am. J. Math. 97, 364-385 (1975; Zbl 0321.57020)] and, more recently, the harmonic Hopf construction, only gave a patchy answer. NEWLINENEWLINENEWLINEIn this paper, the author fills in some of the gaps by showing that harmonic representatives exist for infinitely many elements of \(\pi_{n}({S}^n)=\mathbb{Z}\), \(n \in\mathbb{N}\), the non-trivial element of \(\pi_{n+1}({S}^n)=\mathbb{Z}_{2}\), \(n \geq 3\) and \(\pi_{n+2}({S}^n)=\mathbb{Z}_{2}\), \(n \geq 5\) (odd). NEWLINENEWLINENEWLINEThe idea is to look for maps equivariant under the action of a maximal torus \(({S}^1)^n\) (possibly times \(\mathbb{Z}_{2}\) or \((\mathbb{Z}_{2})^2\), according to dimensions). Existence of a weak equivariant minimiser comes, more or less directly, from the calculus of variation and, if continuous, it is a smooth harmonic map with controlled homotopy class. The crucial point is therefore the regularity of this minimiser \(f\). NEWLINENEWLINENEWLINEThis is reduced to the constancy of all \(H\)-minimising tangent maps of \(f\), where \((\mathbb{R}^n, H)\) is a slice representation of ``lower order'' (\(3 \leq d \leq m\), \(m\) being the dimension of the domain). Lower and upper bounds on the energy of such maps are required to conclude.