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The shape of hyperbolic Dehn surgery space. (English) Zbl 1144.57015
For a compact orientable \(3\)-manifold \(X\) having boundary a single torus, and whose interior admits a hyperbolic structure of finite volume, a celebrated result of Thurston shows that almost all Dehn fillings produce hyperbolizable closed manifolds. This is seen by proving that the hyperbolic Dehn surgery space \(\mathcal{HDS}(X)\), regarded as a subset of a plane whose points correspond to certain incomplete hyperbolic metrics on the interior of \(X\), contains a neighborhood of infinity. The Dehn fillings correspond to the integer lattice points, so all but finitely many produce hyperbolizable manifolds. Thurston also gave a version when \(X\) has more than one boundary torus.
In this paper, continuing previous work in [Ann. Math. (2) 162, No. 1, 367–421 (2005; Zbl 1087.57011)] and other articles, the authors obtain more precise information about \(\mathcal{HDS}(X)\). Writing \(T\) for the boundary torus, \(\mathcal{HDS}(X)\) may be regarded in a natural way as a subset of \(H_1(T;\mathbb{R})\), The first main result states that \(\mathcal{HDS}(X)\) contains the complement of a disk of radius \(7.5832\), centered at the origin. The second is a version of this when \(X\) has more than one boundary torus.
The proofs of these results use deep analytic techniques, which yield a great deal of additional information. For example, for Dehn fillings outside the disk of radius \(7.5832\), the difference in volume between the complete structure on \(X\) and the volume of the filled manifold is at most \(0.198\). More technical results give strong control on the geometry of the manifolds under deformations.
A key technical idea is to work not with the cone manifolds that result from completing the metrics, but with the compact hyperbolic manifolds with boundary that result from truncating the cusps along flat tori. The infinitesimal deformations of such manifolds can be viewed as cohomology classes, and under certain conditions harmonic representatives can be found. One consequence of this is a local rigidity result: under certain hypotheses on the geometry of the boundary tori, there are no infinitesimal deformations keeping the Dehn surgery coefficients constant. This implies that the Dehn surgery coefficients give a local parameterization for cone manifolds. Once the local rigidity and local parameterization theorems are established, the uniform bounds can be deduced using the general approach of the authors’ article mentioned above. The use of hyperbolic manifolds with boundary, as opposed to cone manifolds, produces subtly different estimates in the case of multiple boundary tori.
Unlike the Hodge theory developed in the authors’ previous work, the new version accommodates cone angles greater than \(2\pi\). The resulting harmonic deformation theory has been used by K. Bromberg [J. Am. Math. Soc. 17, No. 4, 783–826 (2004; Zbl 1061.30037), Ann. Math. (2) 166, No. 1, 77–93 (2007; Zbl 1137.30014)] in his proof of the Bers Density Conjecture, as well as by K. Bromberg and J. Brock to obtain more general versions of the Density Conjecture [Acta Math. 192, No. 1, 33–93 (2004; Zbl 1055.57020)].

MSC:
57M50 General geometric structures on low-dimensional manifolds
57N16 Geometric structures on manifolds of high or arbitrary dimension
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