A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds

We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya–Yamaguchi (J. Differ. Geom. 56:1–66, 2000 ; Math. Ann. 333: 131–155, 2005 ) and Morgan–Tian ( arXiv:0809.4040v1 [math.DG], 2008 ). A version of Perelman’s collapsing theorem states: “ Let be a sequence o...

Celý popis

Uložené v:

Podrobná bibliografia
Vydané v:	The Journal of geometric analysis Ročník 21; číslo 4; s. 807 - 869
Hlavní autori:	Cao, Jianguo, Ge, Jian
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer-Verlag 01.10.2011
Predmet:	Abstract Harmonic Analysis Convex and Discrete Geometry Differential Geometry Dynamical Systems and Ergodic Theory Fourier Analysis Global Analysis and Analysis on Manifolds Mathematics Mathematics and Statistics Collapsing Perelman’s critical point theory 53C20 Graph-manifolds Seifert fibered spaces Generalized implicit function theorem
ISSN:	1050-6926, 1559-002X
On-line prístup:	Získať plný text
Tagy:	Pridať tag Žiadne tagy, Buďte prvý, kto otaguje tento záznam!

Popis
Shrnutí:	We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya–Yamaguchi (J. Differ. Geom. 56:1–66, 2000 ; Math. Ann. 333: 131–155, 2005 ) and Morgan–Tian ( arXiv:0809.4040v1 [math.DG], 2008 ). A version of Perelman’s collapsing theorem states: “ Let be a sequence of compact Riemannian 3 -manifolds with curvature bounded from below by (−1) and . Suppose that all unit metric balls in have very small volume, at most v i →0 as i →∞, and suppose that either is closed or has possibly convex incompressible toral boundary. Then must be a graph manifold for sufficiently large i ”. This result can be viewed as an extension of the implicit function theorem. Among other things, we apply Perelman’s critical point theory (i.e., multiple conic singularity theory and his fibration theory) to Alexandrov spaces to construct the desired local Seifert fibration structure on collapsed 3-manifolds. The verification of Perelman’s collapsing theorem is the last step of Perelman’s proof of Thurston’s geometrization conjecture on the classification of 3-manifolds. A version of the geometrization conjecture asserts that any closed 3-manifold admits a piecewise locally homogeneous metric . Our proof of Perelman’s collapsing theorem is accessible to advanced graduate students and non-experts.
ISSN:	1050-6926 1559-002X
DOI:	10.1007/s12220-010-9169-5