Closed minimal surfaces of high Morse index in manifolds of negative curvature

We use Bott’s equivariant Morse theory to show that compact Riemannian three-manifolds with negative sectional curvature possess closed minimal surfaces of arbitrarily high Morse index. 1 Sources of noncompactness for parametrized minimal surfaces In this article, we apply the theory of parametrized two-dimensional minimal surfaces in a compact n-dimensional Riemannian manifold (M, 〈·, ·〉) to the case in which M has strictly negative sectional curvature. We begin with a discussion of how this case fits within the general theory of parametrized minimal surfaces in Riemannian manifolds, as presented in [10]. A parametrized minimal surface of genus g in a curved n-dimensional Riemannian manifold M is a critical point of the two-variable Dirichlet energy E : Map(Σg,M)× Tg → R, where Σg is the compact connected oriented surface of genus g and Tg is the Teichmüller space of marked conformal structures on Σg. This two-variable energy E is defined by the Dirichlet integral,