A remark on regular points of Ricci limit spaces

Abstract

Let Y be a Gromov-Hausdorff limit of complete Riemannian n-manifolds with Ricci curvature bounded from below. A point in Y is called k-regular, if its tangent is unique and is isometric to a k-dimensional Euclidean space. By Cheeger-Colding and Colding-Naber, there is k>0 such that the set of all k-regular point R_k has a full renormalized measure. An open problem is if R_l= ∅ for all l<k? The main result in this paper asserts that if R₁≠∅, then Y is a one-dimensional topological manifold. Our result improves Honda’s result that under the assumption that 1≤dim_H(Y ) <2.

Keywords

Ricci curvature / regular point / Gromov-Hausdorff limit

References

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Higher Education Press and Springer-Verlag Berlin Heidelberg