Harmonic maps form a class of maps between Riemannian manifolds which contains various types of maps of great geometric interest: harmonic functions, geodesics, isometric minimal embeddings and many more. They are critical points of an energy functional and are characterized as solutions of a nonlinear elliptic partial differential equation. We will study existence, uniqueness, stability and regularity questions.
Once existence is established (which is not always the case, however), harmonic maps have important geometric applications. They provide information on the fundamental group, on the isometry group, on certain submanifolds and more.
05Index theory on Lorentzian manifolds15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds