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.

Key words： Ricci curvature; regular point; Gromov-Hausdorff limit

References

1 Cheeger J, Colding T H. Almost rigidity of warped products and the structure of spaces with Ricci curvature bounded below. Ann of Math, 1996, 144(2): 189–237

2 Cheeger J, Colding T H. On the structure of spaces with Ricci curvature bounded below. I. J Differential Geom, 1997, 46: 406–480

3 Cheeger J, Colding T H. On the structure of spaces with Ricci curvature bounded below. II. J Differential Geom, 2000, 54: 13–35

4 Cheeger J, Colding T H. On the structure of spaces with Ricci curvature bounded below. III. J Differential Geom, 2000, 54: 37–74

5 Colding T H, Naber A. Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann of Math, 2012, 176: 1173–1229

6 Colding T H, Naber A. Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications. Geom Funct Anal, 2013, 23(1): 134–148

7 Honda S. On low-dimensional Ricci limit spaces. Nagoya Math J, 2013, 209: 1–22

8 Honda S. Ricci curvature and L^p-convergence. J Reine Angew Math (to appear)

9 Kitabeppu Y, Lakzian S. Characterization of low dimensional RCD^* (K,N) spaces. arXiv: 1505.00420v2 [math. MG], <Date>9 Jun</Date> 2015