Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Collapsing three-dimensional Alexandrov spaces with boundary

Ayato Mitsuishi, Takao Yamaguchi

Abstract

As a continuation of [MY], we determine the topologies of collapsing three-dimensional compact Alexandrov spaces with nonempty boundary.

Collapsing three-dimensional Alexandrov spaces with boundary

Abstract

As a continuation of [MY], we determine the topologies of collapsing three-dimensional compact Alexandrov spaces with nonempty boundary.
Paper Structure (23 sections, 45 theorems, 210 equations, 5 tables)

This paper contains 23 sections, 45 theorems, 210 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. Preliminaries I --- Basics of Alexandrov spaces
  4. Gromov-Hausdorff convergence
  5. Preliminaries II --- Basics of collapsing Alexandrov spaces
  6. Regularities of distance maps
  7. Stability and Fibration Theorems
  8. Rescaling argument
  9. Classification of nonnegatively curved three-spaces with boundaries
  10. Generalized $I$-bundles and generalized Seifert fibrations
  11. Generalized $I$-bundles
  12. Generalized Seifert fibrations
  13. The case that $X$ is a closed surface
  14. The case that $X$ is a surface with boundary
  15. Case (1) -- Inradius collapse
  16. ...and 8 more sections

Key Result

Theorem 1.1

If $0\le \dim X\le 2$, then there is a singular fibration $f_i : M_i \to X$.

Theorems & Definitions (101)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 91 more