Dice Question Streamline Icon: https://streamlinehq.com

Are tangent cone cross-sections Ricci limit spaces?

Determine whether cross-sections of tangent cones of noncollapsed Ricci limit spaces are themselves noncollapsed Ricci limit spaces, beyond being noncollapsed RCD(n−2,n−1) spaces.

Information Square Streamline Icon: https://streamlinehq.com

Background

By combining results of Ketterer and De Philippis–Gigli, the cross-section of each tangent cone of a noncollapsed RCD(K,n) space is known to be a noncollapsed RCD(n−2,n−1) space. For noncollapsed Ricci limit spaces, their tangent cone cross-sections are therefore noncollapsed RCD(n−2,n−1) spaces as well.

However, it is not known whether these cross-sections arise as noncollapsed Ricci limit spaces (i.e., as Gromov–Hausdorff limits of smooth manifolds with uniform lower Ricci and volume bounds). Establishing this would align the tangent cone cross-sections directly with smooth approximation theory.

References

In particular, this is the case for cross-sections of tangent cones of noncollapsed Ricci limit spaces, while such cross-sections are not known to be Ricci limit spaces themselves.