An Overview of the Barycenter Method in Hadamard Spaces 1
Time: 2024-10-16
Published By: Meng Yu
Speaker(s): Shi Wang (Shanghai Tech University)
Time: 15:00-17:00 October 16, 2024
Venue: Room 9, Quan Zhai, BICMR
Abstract: The barycenter method originates from E. Cartan's center-of-mass technique in 1920s, where he uses the convexity of distance function to show the existence and uniqueness of the center-of-mass for Hadamard spaces. In the late 1990s, Besson-Courtois-Gallot followed the similar idea and constructed the celebrated natural map, which proves the minimal entropy rigidity conjecture for rank one symmetric spaces. This leads to many important applications in differential geometry, geometric topology, group theory and dynamical systems.
In the two lectures, we will give a brief introduction to the natural map and all kinds of the applications in the past 30 years. We will discuss examples and their geometry of nonpositively curved spaces, the construction of Patterson-Sullivan measures, the natural map, towards applications to the minimal entropy rigidity theorem, the Mostow's rigidity, the homological vanishing theorems etc. Time permit, we will talk about some open problems in the related fields.
