In 1985, the authors presented a series of lectures on harmonic maps at the University of California at San Diego. Most of these lectures are collected in the present monograph as Part I. Part II contains part of the thesis of the first author and applications to geometry and topology. The final chapter is due to J. Jost and the second author.
Part I is devoted to harmonic maps defined on Riemann surfaces. In the first chapter, general notions and results are stated. Next, Thurston’s and Wolf’s compactifications of Teichmüller spaces are described. Harmonic maps of Kähler manifolds with constant negative holomorphic sectional curvature and minimal surfaces in a Kähler surface are investigated. A characterization of holomorphic immersions into \({\mathbb R}^{2n}\) is proved and applications to the stability of minimal surfaces in \({\mathbb R}^4\) are given. The last chapter is devoted to compact Kähler manifolds of positive bisectional curvature. T. Frankel [Pac. J. Math. 11, 165-174 (1961; Zbl 0107.39002)] conjectured that every compact Kähler manifold of positive holomorphic bisectional curvature is biholomorphic to the complex projective space. The case of dimensions 2 and 3 was proved by Andreotti-Frankel and Mabuchi, respectively. Using harmonic maps, Y.-T. Siu and S.-T. Yau [Invent. Math. 59, 189-204 (1980; Zbl 0442.53056)] proved the Frankel conjecture for arbitrary dimension.
The theory of harmonic maps defined on a higher-dimensional manifold is based on the work of J. Eells and J. H. Sampson [Am. Math. J. 86, 109-160 (1964; Zbl 0122.40102)]. Instead of the variational approach, they used a heat equation argument, which has deep influence in geometry.
The first two chapters of Part II deal with the regularity theory even when the target manifold is not necessarily negatively curved. This uses the theory of Sobolev spaces. Finally, very interesting applications to geometry and topology are obtained. For instance, compact groups acting on a compact manifold are studied and some classification theorems in this respect are proved. Also, the authors focus on the topology of stable hypersurfaces in manifolds with non-negative Ricci curvature. The following obstruction theorem is proved. Let \(M\) be a complete non-compact stably immersed hypersurface in a manifold of non-negative curvature and \(D\) a compact domain in \(M\) with smooth, simply connected boundary. Then there is no non-trivial homomorphism from \(\pi_1(D)\) into the fundamental group of a compact manifold with non-positive curvature.
The monograph under review covers a series of recent publications on harmonic maps and their relationship with geometry and topology of Teichmüller spaces and Kähler manifolds. Consequently, it is a fundamental reference for those working on such subjects.