Noncompact harmonic manifolds

TL;DR

This paper surveys recent advances and presents new results on noncompact simply connected harmonic manifolds, exploring their classification and properties, especially those with purely exponential volume growth.

Contribution

It offers a comprehensive survey of recent findings and introduces new results on the structure and classification of noncompact harmonic manifolds.

Findings

Many new results for noncompact harmonic manifolds.

Classification insights for manifolds with exponential volume growth.

Clarification of the landscape beyond the Lichnerowicz conjecture.

Abstract

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. Szabo for harmonic manifolds with compact universal cover. E. Damek and F. Ricci 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 noncompact 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.