New Examples of Collapse With Lower Curvature Bound
In this work, we describe a method to construct new examples of collapse with a lower curvature bound inspired by Cheeger and Gromov. Unlike with collapse with an upper and lower curvature bound, which is now completely understood, the structure of collapse with a lower curvature bound is still a mystery.
In , Grove and Petersen showed that if a sequence of Riemannian manifolds (M i , g i ) has uniform lower curvature bound, k, and M i → X, then X is an Alexandrov space with lower curvature bound k. Petersen, Wilhelm, and Zhu then showed that the converse is false . Perelman showed that given a sequence of n-dimensional Alexandrov spaces with a uniform lower curvature bound, with limit space X such that dim X = n, all but finitely many of the prelimits are homeomorphic to X.
In 1985, Cheeger and Gromov introduced the concept of an F-structure, which can be thought as a generalized torus action on a manifold . They showed that a manifold collapses with bounded curvature if and only if it admits an F-structure . An F-structure is an example of a more general construct known as a g-structure, which is a sheaf of Lie groups actions, and is one of the main tools used in our approach.
The second main tool we use was also defined by Cheeger and is known as a Cheeger deformation. Cheeger generalized the method used by Berger in his classic example now known as the Berger spheres, which helped create the study of collapse in Riemannian geometry. Berger showed that scaling the metric of S 3 along the Hopf circles collapses S 3 to the 2-sphere of radius 1/2.
In this work, using g-structures and Cheeger deformations, we construct new examples of collapse with a lower curvature bound.