Gromov-Hausdorff limit of orthonormal frame bundles of non-collapsed manifolds with bounded Ricci curvature

TL;DR

This paper investigates the Gromov-Hausdorff limits of orthonormal frame bundles over non-collapsed manifolds with bounded Ricci curvature, revealing structural similarities to Ricci limit spaces.

Contribution

It establishes that the limit of these frame bundles retains key geometric properties, including singular set codimension and regularity of the regular part.

Findings

Limit space has singular set of codimension at least 4.

Regular part of the limit space contains an open dense $C^{1,\alpha}$-Riemannian manifold.

The limit space shares properties with Ricci limit spaces of non-collapsing sequences.

Abstract

Let $M_i$ be a sequence of non-collapsed $n$-manifolds with two-sidedly bounded Ricci curvature. We show that the Gromov-Haudorff limit space, $Y$, of the associated sequence of orthonormal frame bundles, $FM_i$, equipped with an almost canonical metric, shares similar properties as a Ricci limit space of non-collapsing sequence i.e., the singular set has codimension $\ge 4$ whose complement contains an open and dense $C^{1,\alpha}$-Riemannian manifold.