In this note, we show that some F -harmonic maps into spheres are global maxima of the variations of their energy functional on the conformal group of the sphere. Our result extends partially those obtained in [15] and [17] for harmonic and p-harmonic maps.

Abstract For the Schrödinger flow from R × R to the 2-sphere S, it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic maps until the blow up time (if any), and that they blow up if and only if the length scale of th...

For a bounded domain Ω equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from (Ω, g) to a compact Riemannian manifold (N, h) ↪→ Rk without boundary. We generalize the notion of stationary harmonic maps and prove their partial regularity. We also discuss the global Lipschitz and piecewise C1,α-regularity of harmonic maps from (Ω, g) to manifolds that su...

We prove that a quasi-isometric map, and more generally coarse embedding, between pinched Hadamard manifolds is within bounded distance from unique harmonic map.

We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal surfaces. In the genus 0 case, our flow is just the harmonic map flow, and it tries to find branched minimal 2-spheres as in Sacks-Uhlenbeck [13] and Struwe [16...