Noncompact harmonic manifolds
MFO Scientific ProgramResearch in Pairs 2012
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The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z.I. Szab ́o [Sz] for harmonic manifolds with compact universal cover. E. Damek and F. Ricci [DR]provided examples showing that in the noncompact case the conjecture is wrong. However, such manifolds do not admit a compact quotient. The classification of all noncompact harmonic spaces is still a very difficult open problem. In this paper we provide a survey on recent results on non-compact simply connected harmonic manifolds, and we also prove many new results, both for general noncompact harmonic manifolds and for noncompact harmonic manifolds with purely exponential volume growth.