×

Perelman’s stability theorem. (English) Zbl 1151.53038

Cheeger, Jeffrey (ed.) et al., Metric and comparison geometry. Surveys in differential geometry. Vol. XI. Somerville, MA: International Press (ISBN 978-1-57146-117-9/hbk). Surveys in Differential Geometry 11, 103-136 (2007).
The author provides a comprehensive and detailed exposition for the celebrated stability theorem of G. Perelman [Aleksandrov spaces with curvatures bounded from below II, preprint (1991)]: if \(X^n_i\) is a sequence of compact \(n\)-dimensional Aleksandrov spaces of curvature \(\geq \kappa\) and diameter \(\leq D\) converging in Gromov-Hausdorff topology to a space \(X\) of dimension \(n\), what means that a sequence of Hausdorff approximations \(f_i:X^n_i\to X\) was fixed, then \(f_i:X^n_i\to X\) can be approximated by homeomorphisms for all large \(i\). As was pointed out by G. Ya. Perelman, his original proof can be simplified using the constructions developed in two of his later papers [St. Petersbg. Math. J. 5, No. 1, 205–213 (1993; Zbl 0815.53072), “DC structure on Aleksandrov space with curvature bounded below”, preprint (1995; http://www.math.psu.edu/petrunin/papers/alexandrov/Cstructure.pdf)].
In the present paper, the author carries out these simplifications. One of the main ingredients of the proof is the Morse theory for functions on Aleksandrov spaces and Perelman’s constructions [Zbl 0815.53072], in particular the fibration theorem, which implies that Aleksandrov spaces are stratified topological manifolds. Another important tool in the proof of stability is the gluing theorem derived from deformations of homeomorphisms results of L. C. Siebenmann [Comment. Math. Helv. 47, 123–163 (1972; Zbl 0252.57012)], which allows to reduce the stability theorem to the local situation. To construct local stability homeomorphisms near a point \(p\in X\) one argues by reverse induction on the dimension \(k\) of the strata containing \(p\). The base of induction follows from the fact that a map \(f:X^n\to \mathbb{R}^n\) is a local homeomorphism near a regular point, which is a consequence of the above mentioned fibration theorem.
An important role in the induction step is played by a technical construction of strictly concave functions obtained by manipulating distance functions [see G. Perelman, Zbl 0815.53072]. One should note that the proof of the stability theorem requires to prove a stronger version of it, namely a parameterized one (Theorem 7.8), that allows to deduce a finiteness theorem for submetries (for basic information on submetries, see [A. Lytchak, Allgemeine Theorie der Submetrien und verwandte mathematische Probleme, Bonner Mathematische Schriften. 347. Bonn: Univ. Bonn, Mathematisches Institut (Thesis 2001) (2002; Zbl 1020.54021)]. In particular, this parameterized stability theorem immediately implies a finiteness theorem for Riemannian submersions proved by K. Tapp [Proc. Am. Math. Soc. 130, No. 6, 1809–1817 (2002; Zbl 0997.53028), Theorem 2], which is a generalization of a result due to K. Grove, P. V. Petersen J.-Y. Wu [Invent. Math. 99, No. 1, 205–213 (1990; Zbl 0747.53033)].
The author generalizes the stability theorem by showing that the stability homeomorphisms can be chosen to preserve stratification of Aleksandrov spaces into extremal subsets [for the notion of an extremal subset in an Aleksandrov space, see G. Perelman and A. Petrunin [St. Petersbg. Math. J. 5, No. 1, 215–227 (1994); translation from Algebra Anal. 5, No.1, 242–256 (1993; Zbl 0802.53019)].
For the entire collection see [Zbl 1149.53005].

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C20 Global Riemannian geometry, including pinching
PDFBibTeX XMLCite
Full Text: arXiv