Geometry of Harmonic Maps

A structure theorem of Dirac-harmonic maps between spheres

“…We refer to [4][5][6] for more background material on spin structures and Dirac operators and to [7,8] for more background material on harmonic maps.…”

Some Remarks on Transversally Harmonic Maps

“…In this section, we apply the results of Xin on Riemannian submersions, cf. [39], to obtain various characterisations of transversally harmonic maps.…”

Tension Field and Index Form of Energy-Type Functionals

“…To that end let v be the gradient vectorfield on S n ⊂ ‫ޒ‬ n+1 of the restriction of a linear map p : ‫ޒ‬ n+1 → ‫,ޒ‬ p(x) = p, x for a unit vector p ∈ ‫ޒ‬ n+1 as in [9]. Then v(x) 2 + p(x) 2 = 1 and ∇ x v = −px for all x ∈ TS n , hence ∇ e i v, e j = −pδ i,j .…”

“…The Bochner formula for vector fields then implies that isometries are -stable under certain conditions on the first and second derivative of . As in the classical case, (see [4], [9]) there is also a range of maps such that the identity on the sphere S n is unstable for the -energy.…”

“…the survey papers [5,6]. For a discussion of stability results in this case we refer to [9] and the references there.…”