1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

CHAPTER 31

Hyperbolic Geometry and 3-Manifolds

We've got no future, we've got no past

Here today, built to last.

• From "West End Girls" by Pet Shop Boys

The purpose of this ch apter is to prepare the reader for the discussion of nonsin-

gular solutions to the Ricci flow on closed 3 -manifolds in the subsequent chapters.

The main aspect of nonsingular solutions related to t his chapter is the occurren ce

therein of finite-volume hyperbolic limits and their corresponding pieces in the so-

lutions.

In §1 we review basic facts about hyperbolic space, its isometries, and its quo-

t ients. In §2 we recall some basic geometric and topological facts related to hyper-

bolic manifolds. In §3 we discuss the Margulis lemma and ends of finite-volume

hyperbolic manifolds. In §4 we recall the Mostow rigidity theo rem. In §5 we dis-

cuss Seifert fibered manifolds and graph manifolds in relation to Cheeger-Gromov

collapse.

1. Introduction to hyperbolic space

In this section we discuss models and isometries of hyperbolic space.

1.1. Models of hyperbolic n-space !Hln.

Let !Hln, n 2: 2, denote hyperbolic n-space, the simply-connected complete

Riemannian n-manifold with constant sectional curvature equ al to -1. By the

Cartan- Ambrose- Hicks theorem, hyperbolic n-space is unique up to isometry.

There are various concrete models of hyperbolic n-space; we note three of them.

The disk model is

with the metric

. 4 ( dxi + .. · + dx;,)
9 1Il> =:= (1 - lxl2)2

The ideal boundary cWn ~ { x : !xi = 1} is also called the sphere at infinity.

The upper half-space model is un ~ {x E ]Rn: Xn > O} with the metric

. dxi + · .. + dx;,
gQJ =:= 2
Xn

We call the ideal boundary aun ~ {x: Xn = O} the hyperplane at infinity.

With its standard co nformal structure, the one point compactification of aun by

the point at infinity is conformally equi valent to the sphere at infinity.

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