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

TL;DR

This paper constructs the first known example of an open manifold with positive Ricci curvature that has a non-proper Busemann function, addressing a longstanding open question in differential geometry.

Contribution

It provides the first explicit example of an open manifold with positive Ricci curvature exhibiting a non-proper Busemann function, challenging previous assumptions.

Findings

First example of such a manifold with positive Ricci curvature

Counterexample to the properness of Busemann functions in this setting

Addresses a longstanding open question in geometry

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.