Collapsing Geometry with Ricci Curvature Bounded Below and Ricci Flow Smoothing

2020/08/31 by Shaosai Huang, Xiaochun Rong, Bing Wang
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Bounded function #Curvature #Curvature of Riemannian manifolds #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Pure mathematics #Ricci curvature #Ricci flow #Riemann curvature tensor #Scalar curvature #Sectional curvature #math.DG #msc:53C21 #msc:53C23 #msc:53E20

paper · pdf · doi:10.3842/sigma.2020.123

published as SIGMA 16 (2020), 123, 25 pages · 25 pages, dedicated to Misha Gromov on the occasion of his 75th birthday

arxiv created 2020/11/30 · openalex publication_date 2020/11/30 · arxiv updated 2020/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• The splitting theorem for manifolds of nonnegative Ricci curvature
• Gromov–Hausdorff collapsing of Calabi–Yau manifolds
• Three-manifolds with positive Ricci curvature
• Large Complex Structure Limits of K3 Surfaces
• Deforming the metric on complete Riemannian manifolds
• Geometric Structures of Collapsing Riemannian Manifolds I
• Mirror symmetry is T-duality
• Collapsing of Riemannian manifolds and eigenvalues of Laplace operator
• The fundamental equations of a submersion.
• Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications
• Space of Ricci flows (II)
• Short-time existence of the Ricci flow on complete, non-collapsed 3-manifolds with Ricci curvature bounded from below
• Quantitative Volume Space From Rigidity with lower Ricci curvature bound II
• The local entropy along Ricci flow---Part A: the no-local-collapsing theorems
• Rectifiability of Singular Sets in Noncollapsed Spaces with Ricci Curvature bounded below
• Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces
• A New Proof of the Gromov's Theorem on Almost Flat Manifolds
• Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications
• On the long-time behavior of immortal Ricci flows
• SYZ conjecture for Calabi-Yau hypersurfaces in the Fermat family
• Rigidity of the first Betti number via Ricci flow smoothing