Constructions of harmonic maps between Hadamard manifolds
This thesis presents several methods of constructing harmonic maps between Hadamard manifolds. The first method is motivated by \textit{R. T. Smith}'s join construction [Am. J. Math. 97, 364-385 (1975; Zbl 0321.57020)] of harmonic maps between spheres. A similar construction is defined, which reduces the equation of harmonic maps between real hyperbolic spaces to an ordinary differential equation. A large portion of the thesis is devoted to the study of such o.d.e.\ in rather general form. As a result of solving this o.d.e., it is proved that full harmonic maps between real hyperbolic spaces \(RH^m\to RH^n\) exist for many values of \((m,n)\). NEWLINENEWLINENEWLINESince it is essential for this construction to have harmonic eigenmaps (i.e.\ harmonic maps between spheres of constant energy density) to start with, a separate section about harmonic eigenmaps is included. In this section, which is interesting in its own right, the author studies systematically how to construct full eigenmaps of polynomial degree two. This has direct consequences for the existence theorems mentioned above. NEWLINENEWLINENEWLINEFurthermore, following an idea of \textit{H. Donnelly} [Trans. Am. Math. Soc. 344, No. 2, 713-735 (1994; Zbl 0812.58020)], an existence and uniqueness result is established for proper harmonic maps between Damek-Ricci spaces, which are a generalization of rank one symmetric spaces of noncompact type. NEWLINENEWLINENEWLINEIn the last section, there is a nonexistence result for proper harmonic maps from complex hyperbolic space to real hyperbolic space. If the dimensions are at least two, then there is no proper harmonic map of class \(C^2\) which has \(C^1\)-regularity up to the ideal boundary.
scientific article; zbMATH DE number 1680253