Manchester Geometry Seminar 1999/2000

Harmonic maps f:M ® N between smooth Riemannian manifolds are the extremals (= solutions of the Euler-Lagrange equations) of the energy functional E(f) = 1/2 (\integral |d(f)|²). Examples: If M is the circle, then the closed geodesics of N are harmonic. If N is the circle, then the abelian integrals are harmonic. That theory is extended to maps f:X ® Y between Riemannian polyhedra. For instance, between normal complex analytic spaces; or spaces with conical singularities. That extension is conceptually different from the smooth case, requiring quite different ideas.