Topology and 𝜖–regularity theorems on collapsed manifolds with Ricci curvature bounds

2014/12/31 by Aaron Naber, Ruobing Zhang
Mathematics · #Advanced Operator Algebra Research #Ball (mathematics) #Curvature #Einstein #Geometric Analysis and Curvature Flows #Manifold (fluid mechanics) #Nilpotent #Nonlinear Partial Differential Equations #Ricci curvature #Ricci-flat manifold #Topology (electrical circuits) #math.DG

paper · pdf · doi:10.2140/gt.2016.20.2575

published as Geom. Topol. 20 (2016) 2575-2664

arxiv created 2015/01/05 · openalex created_date 2016/06/24 · openalex publication_date 2016/10/07 · arxiv updated 2016/10/19 · openalex updated_date 2026/08/05

Abstract

In this paper we discuss and prove [math] –regularity theorems for Einstein manifolds [math] , and more generally manifolds with just bounded Ricci curvature, in the collapsed setting. ¶ A key tool in the regularity theory of noncollapsed Einstein manifolds is the following. If [math] is such that [math] and that [math] is sufficiently Gromov–Hausdorff close to a cone space [math] for [math] , then in fact [math] on [math] . No such results are known in the collapsed setting, and in fact it is easy to see that without further assumptions such results are false. It turns out that the failure of such an estimate is related to topology. Our main theorem is that for the above setting in the collapsed context, either the curvature is bounded, or there are topological constraints on [math] . ¶ More precisely, using established techniques one can see there exists [math] such that if [math] is an Einstein manifold and [math] is [math] –Gromov–Hausdorff close to ball in [math] , then the fibered fundamental group [math] is almost nilpotent with [math] . The main result of the this paper states that if [math] is maximal, then [math] on [math] . In the case when the ball is close to Euclidean, this is both a necessary and sufficient condition. There are generalizations of this result to bounded Ricci curvature and even just lower Ricci curvature.

Citations

• The splitting theorem for manifolds of nonnegative Ricci curvature
• Large Complex Structure Limits of K3 Surfaces
• Regularity of Einstein manifolds and the codimension 4 conjecture
• Differential equations on riemannian manifolds and their geometric applications
• Convergence and rigidity of manifolds under Ricci curvature bounds
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications