The Morse theory of two-dimensional closed branched minimal surfaces and their generic non-degeneracy in Riemannian manifolds
After some informal history and some Teichmüller and harmonic mapping theory of closed surfaces, following the exposition of \textit{A. Tromba} [``Teichmüller Theory in Riemannian Geometry'', Birkhäuser, Basel (1992; Zbl 0785.53001)], the author consider maps from a compact Riemann surface of genus \(\geq 2\) to a compact Riemannian manifold \((N,G)\) of strictly negative curvature. In that situation (essential uniqueness of harmonic representatives), the energy functional can be adjusted to give an energy \(\widetilde E\) whose critical points are the conformal harmonic maps. In a homotopy class \(\Sigma\) whose elements induce isomorphisms of fundamental groups, \(\widetilde E\) is proper -- so classical Morse theory applies. Much of this paper is devoted to generic non-degeneracy with respect to the metric \(G\) on \(N\), based on [\textit{A. Tromba}, J. Functional Anal. 23, 362-368 (1976; Zbl 0341.58005)]. Here the homomorphisms of the fundamental groups induced by the maps of \(\Sigma\) are assumed injective.
scientific article; zbMATH DE number 1454749