3-dimensional asymptotically harmonic manifolds with minimal horospheres (Q2634773)

A manifold is called \textit{asymptotically harmonic} if the mean curvature of its horospheres is a constant \(h>0\). In this paper the author studies 3-dimensional, complete, simply connected, asymptotically harmonic manifolds \((M,g)\) without conjugate points, showing that \(h = 0\) implies \((M,g)\) to be a flat manifold. This result complements earlier results, he achieved jointly with \textit{V. Schroeder} establishing under the same assumptions that \(h > 0\) implies \((M,g)\) to be hyperbolic [ibid. 90, No. 3, 275--278 (2008; Zbl 1137.53010)].