Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 123, 25 pages      arXiv:2008.12419      https://doi.org/10.3842/SIGMA.2020.123
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Collapsing Geometry with Ricci Curvature Bounded Below and Ricci Flow Smoothing

Shaosai Huang a, Xiaochun Rong b and 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

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

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

Key words: almost flat manifold; collapsing geometry; locally bounded Ricci covering geometry; nilpotent Killing structure; Ricci flow.

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