The Moduli Space of Hyperbolic Cone Structures

Introduction. Roughly speaking, a cone structure is a manifold together with a link each of whose component has a cone angle attached. It is a kind of singular manifold structure. If each cone angle is of the form 2π/n, for some integer n, the cone structure becomes an orbifold structure. Unlike an orbifold structure, the cone structure is not a natural concept, but it turns out to be very important in the study of geometric orbifolds. In this paper, we will consider 3-dimensional geometric cone structures. The main result in this paper is an existence and uniqueness theorem for 3-dimensional hyperbolic cone structures. Theorem A. Let Σ be a hyperbolic link with m components in a 3-dimensional manifold X. Then the moduli space of marked hyperbolic cone structures on the pair (X,Σ) with all cone angles less than 2π/3 is an m-dimensional open cube, parameterized naturally by the m cone angles. Theorem A is an analogue of Mostow’s rigidity theorem. If we have two hyperbolic cone structures C1 and C2 with cone angles less than 2π/3, then C1 and C2 are isometric if and only if there is a homeomorphism between (X1,Σ1) and (X2,Σ2) so that corresponding cone angles are the same. The proof of the theorem goes in a similar way as Thurston proposed for the proof of his geometrization theorem for orbifolds [T2]. As a corollary, we will give a proof of the following special case of Thurston’s geometrization theorem. Corollary B. If M is an irreducible, closed, atoroidal 3-manifolds, it is not Seifert manifold and admits a finite group G action. If the order of G is odd, the G-action is effective and not fixed-point-free, then the quotient M/G is a geometric orbifold. This paper is orgnized in seven sections. §1 collects some basic facts for geometric cone manifolds and their limits, we refer reader to somewhere else for details. Our first goal is using equivariant Ricci flow to study the topology and geometry of compact Euclidean cone manifolds. A briefly review of Hamilton’s work on Ricci flow in given in §2, and we also establish a version of Ricci flow on orbifold. Using Ricci flow on orbifold