Shaosai Huang, Xiaochun Rong, Bing Wang, “Collapsing Geometry with Ricci Curvature Bounded Below and Ricci Flow Smoothing”, SIGMA, 2020, Volume 16,123, 25 pp.

This article is cited in 2 papers

Collapsing Geometry with Ricci Curvature Bounded Below and Ricci Flow Smoothing

Shaosai Huang^a, Xiaochun Rong^b, Bing Wang^c

^a Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA
^b Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA
^c Institute of Geometry and Physics, and School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui Province, 230026, China

Abstract: 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 metric together with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.

Keywords: almost flat manifold, collapsing geometry, locally bounded Ricci covering geometry, nilpotent Killing structure, Ricci flow.

MSC: 53C21, 53C23, 53E20

Received: August 30, 2020; in final form November 23, 2020; Published online November 30, 2020

Language: English

DOI: 10.3842/SIGMA.2020.123