Initial-final value problems for ordinary differential equations and applications to equivariant harmonic maps

摘要

It is a fundamental problem to show the existence or nonexistence of a harmonic map between complete Riemannian manifolds. In the case of compact manifolds, a remarkable existence result is due to Eells-Sampson. They showed that there exists a harmonic map if the target manifold has nonpositive sectional curvature. However, there is no general theory in the case the target manifold has positive sectional curvature. As for spheres, Smith reduced the harmonic map equation to an ordinary differential equation and solving it, he constructed harmonic maps between spheres (see [2, 12, 13] for details and related topics).