Normalized Harmonic Map Heat Flow

Abstract

For any closed Riemannian manifold N we propose the normalized harmonic map heat flow as a means to obtain nonconstant harmonic maps uW S m ! N , m 3.Finding nonconstant harmonic maps uW S m ! N R n for a closed target manifold N and m 3 is a prototype of a supercritical variational problem.El Soufi [12] showed that any nontrivial (sufficiently smooth) harmonic map uW S m ! N either achieves a strict maximum of the Dirichlet energywith respect to the action 3 ! u ı of the Möbius group of S m on u, or is constant in the direction defined by some a 2 S m . Thus, given a smooth map u 0 W S m ! N , one might hope to find a harmonic map homotopic to u 0 as a critical point of D of mountain-pass type, minimizing the quantity sup 2 D.uı / among maps u homotopic to u 0 . Even though this min-max procedure at first may look similar to the study of sweepouts in the work of Marques-Neves [22] on the Willmore conjecture, the fact that in our case the Möbius group acts from the right makes a decisive difference. Indeed, in our setting for any u each 2 induces a "variation of the independent variables" while keeping the image of u fixed, whereas in the setting of Marques-Neves the map u itself is deformed. A further complication arises from the fact that when applying the harmonic map heat flow to -type sweepouts of the form .u ı / 2 , singularities arise instaneously, possibly destroying any topological data. Moreover, since the heat flow in general does not commute with 2 , the family of such -type sweepouts is not invariant under the flow; that is, flowing a -type sweepout for any time t > 0 in general will fail to produce a -type sweepout.

Rigidity of the Hopf fibration

“…If τ = 0 on U , then (41) gives that the function ψ is constant on U . Using (47), we obtain that z = 0 on U . In the sequel, we consider the open subset…”

“…However, the existence of non constant harmonic maps between Communicated by J. Jost. Riemannian manifolds, especially between spheres, is a highly non trivial issue; see for example [4,13,14,31,46,47,53].…”