Collapsing Geometry with Ricci Curvature Bounded Below and Ricci Flow Smoothing

Shaosai Huang, University of Wisconsin-Madison, USA, Xiaochun Rong, Bing Wang, Rutgers University, USA, University of Science and Technology of China, China
2020 Symmetry, Integrability and Geometry: Methods and Applications  
We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kähler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.
doi:10.3842/sigma.2020.123 fatcat:ncd6mxdp6bbdlp3mxm2g2btr7e