Examples of open manifolds with positive Ricci curvature and non-proper Busemann functions

Abstract

We give the first example of an open manifold with positive Ricci curvature and a non-proper Busemann function at a point. This provides counterexamples to a longtime well-known open question whether the Busemann function at a point of an open manifold with nonnegative Ricci curvature is proper.

arXiv commentAdded some discussions on the Busemann functions defined by geodesic spheres