This paper gathers together some applications of quasigeodesic and gradient curves. After a discussion of extremal subsets, we give a proof of the Gluing Theorem for multidimensional Alexandrov spaces, and a proof of the Radius Sphere Theorem.

This paper can be considered as a continuation of [Perelman and Petrunin 1994]. It gathers together some applications of quasigeodesic and gradient curves. The first section considers extremal subsets; in the second section we prove the Gluing Theorem for multidimensional Alexandrov spaces; in the third we give another proof of the Radius Sphere Theorem. Our terminology and notation are those of [Perelman and Petrunin 1994] and [Burago et al. 1992]. We usually formulate the results for general Alexandrov space, but for simplicity give proofs only for nonnegative curvature.

1. Intrinsic Metric of Extremal Subsets

The notion of an extremal subset was introduced in [Perelman and Petrunin 1993, 1.1], and has turned out to be very important for the geometry of Alexandrov spaces. It gives a natural stratification of an Alexandrov space into open topological manifolds. Also, as is shown in recent results of G Perelman, extremal subsets in some sense account for the singular behavior of collapse. Therefore the intrinsic metric of such subsets turns out to be important. Moreover, there is hope that extremal subsets with intrinsic metric will give a way to approach the idea of multidimensional generalized spaces with bounded integral curvature.

In this section we give a new proof of the generalized Lieberman lemma, prove a kind of “stability” property for extremal subsets, and prove the first variation formula for the intrinsic metric of extremal subsets.