Abstract
About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of hyperbolic geometry, as natural as Euclid’s regular polyhedra. For a closed manifold, Mostow rigidity assures that a hyperbolic structure is unique when it exists [Mos], so topology and geometry mesh harmoniously in dimension 3. This remarkable theorem applies to all 3-manifolds which can be built up in an inductive way from 3-balls, i.e. Haken manifolds. Thurston’s construction of a hyperbolic structure is also inductive. At the inductive step one must find the right geometry on an open 3-manifold so that its ends may be glued together. Using quasiconformal deformations, the gluing problem can be formulated as a fixed-point problem for a map of Teichmuller space to itself. Thurston proposes to find the fixed point by iterating this map. Here we outline Thurston’s construction, and sketch a new proof that the iteration converges. Our argument rests on a result entirely in the theory of Riemann surfaces: an extremal quasiconformal mapping can be relaxed (isotoped to a map of lesser dilatation) when lifted to a sufficiently
Highlights
About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature
The gluing problem can be formulated as a fixed-point problem for a map of Teichmüller space to itself
Our argument rests on a result entirely in the theory of Riemann surfaces: an extremal quasiconformal mapping can be relaxed when lifted to a sufficiently large covering space
Summary
About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. Mostow rigidity assures that a hyperbolic structure is unique when it exists [Mos], so topology and geometry mesh harmoniously in dimension 3 This remarkable theorem applies to all 3-manifolds, which can be built up in an inductive way from 3-balls, i.e., Haken manifolds. Our argument rests on a result entirely in the theory of Riemann surfaces: an extremal quasiconformal mapping can be relaxed (isotoped to a map of lesser dilatation) when lifted to a sufficiently large covering space (e.g., the universal cover). This contraction gives an immediate estimate for the contraction of Thurston's iteration. Background in Teichmüller theory can be found, e.g., in [Bers[2], Bers[4], Gard]
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