Compactifications of Complete Riemannian Manifolds and Their Applications

To study a noncompact Riemannian manifold, it is often useful to …nd a compacti…cation or attach a boundary. For example, in hyperbolic geometry a lot of investigation is carried out on the sphere at in…nity. An eminent illustration is Mostow’s proof of his rigidity theorem for hyperbolic manifolds [Mo]. More generally, if f M is simply connected and nonpositively curved, one can compactify it by equivalent geodesic rays and the boundary is a topological sphere, called the geometric boundary. This compacti…cation was …rst introduced in [EO] and has been indispensable in the study of negatively curved manifolds. If f M is not nonpositively curved, then the geometric compacti…cation does not work in general. But there are other compacti…cations which are useful for various studies. In this short survey, we will discuss some of these compactifcations and the relationships among them. Our discussion will focus on general Riemannian manifolds and therefore we ignore the large literature on compacti…cations of symmetric spaces (see the book [GJL]). We …rst discuss the geometric compacti…cation for Cartan-Hadamard manifolds and Gromov hyperbolic spaces in Section 2. In Section 3 we discuss the Martin compacti…cation. In Section 4 we discuss the Busemann compacti…cation. In the last section, we discuss how these compacti…cations are used. In particular, we consider certain invariants de…ned on the Martin boundary and prove a comparison inequality using a method of Besson, Courtois and Gallot. It should be noted that when the author showed this inequality to François Ledrappier he was informed that it had been known to Besson, Courtois and Gallot (unpublished).