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.

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