The Grushin hemisphere as a Ricci limit space with curvature $\ge 1$

TL;DR

This paper constructs a sequence of Riemannian metrics on spheres with Ricci curvature at least 1, whose Gromov-Hausdorff limit is the Grushin hemisphere, illustrating a Ricci limit space with degeneracy along an equator.

Contribution

It demonstrates how to realize the Grushin hemisphere as a Ricci limit space with curvature bounds, expanding understanding of degenerations in Riemannian geometry.

Findings

The sequence of metrics converges to the Grushin hemisphere in the Gromov-Hausdorff sense.

The constructed metrics maintain Ricci curvature $\,\ge 1$ throughout the sequence.

The work provides a new example of Ricci limit spaces with degeneracies along a submanifold.

Abstract

The Grushin sphere is an almost-Riemannian manifold that degenerates along its equator. We construct a sequence of Riemannian metrics on a sphere $S^{m+n}$ with $Ric\ge 1$ such that its Gromov-Hausdorff limit is the $n$-dimensional Grushin hemisphere.