Some of my work has involved energy-minimizing harmonic mappings between manifolds, in the case where discontinuities occur. For example, the radial projection from the (n+1) - dimensional unit ball to the n - dimensional sphere has minimum p-energy among maps to the n - sphere with the same Dirichlet boundary values if 1 <= p <= n and p is an integer. (Real values of p < n - 1 remained an open problem until a recent breakthrough by Min-chun Hong.) For another example, the Hopf maps from the (2n - 1) - dimensional sphere to the n - dimensional sphere, defined using the complex numbers, quaternions or Cayley numbers, when extended to be positive homogeneous of degree zero and restricted to the 2n - dimensional unit ball, n = 2, 4, 8 resp., have minimum energy for their Dirichlet boundary values among maps into the n - sphere (all joint with Jean-Michel Coron [31]). We are still very far from having a complete catalogue of such minimizing tangent maps. |
[31]. Minimizing p-Harmonic Maps into Spheres (with Jean-Michel Coron), J. Reine Angewandte Math., 401, 82-100 (1989).