Equivariant Harmonic Maps

Abstract

Harmonic maps between Riemannian manifolds satisfy a system of quasi-linear partial differential equations. In order to have existence results one would solve PDE’s on certain manifolds. In the case when the sectional curvature of the target manifold is nonpositive or the image of the map is contained in a geodesic convex neighborhood, such a problem has been solved in [E-S], [H-K-W] and [S-U1] by PDE method. But, for maps into positively curved manifolds, especially for harmonic maps between spheres, the PDE method is not successful. By heat flow method the solution blows up at finite time (cf. [C-G], [D2], [C-D]). In this case any harmonic map is unstable (see [X1], [Le1]) and the direct method is not applicable. If the manifolds possess an additional geometric structure and the maps between them are equivariant, one can find special solutions to the harmonicity equations. This method has been successfully utilized in [Sm2], [P-R], [D1], [E-R1] [Ur], [X13], [X14] and [X15]. Recently, in their monograph [E-R2] Eells-Ratto emphasize the ODE method to the elliptic variational problems. The present chapter is also devoted to the equivariant harmonic maps. Besides single ODE, the reduction equations can also be a system of ODE’s or a single PDE with fewer independent variables than the original harmonicity equations. Such examples show that the equivariant method is a potential method to find nonminimal critical points in geometric variational problems.