Pure Maths Colloquium: A survey on non-compact harmonic manifolds
16 November 2015 16:00 in CM221
A complete Riemannian manifolds is called harmonic iff harmonic functions have the mean value property, i.e. the average of harmonic functions over a geodesic sphere coincide with it’s value at the center. In 1944 Lichnerowicz conjectured that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. In 1990 the conjecture has been proved by Z. Szabo for harmonic manifolds with compact universal cover. Furthermore, the conjecture was obtained by Besson, Courtois and Gallot for compact manifolds of strictly negative curvature as an application of their entropy rigidity theorem in combination with the rigidity theorems by Benoist, Foulon and Labourie on stable and unstable foliations.
On the other hand, E. Damek and F. Ricci provided examples showing that in the non-compact case the conjecture is wrong. However, such manifolds do not admit a compact quotient.
In this talk we will present recent results on simply connected, non-compact and non-flat harmonic spaces. In particular for such spaces X the following properties are equivalent: X has rank 1, X has purely exponential volume growth, X is Gromov hyperbolic, the geodesic flow on X is Anosov with respect to the Sasaki metric.
Furthermore we obtain, that no focal points imply the above properties. Combining those results with the above mentioned rigidity theorems shows that the Lichnerowicz conjecture is true for all compact harmonic manifolds without focal points or with Gromov hyperbolic fundamental groups. Some of the results have been generalized in collaboration with Norbert Peyerimhoff to asymptotically harmonic manifolds which we briefly mention if time permits.
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