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Teichmüller geodesics and ends of hyperbolic 3-manifolds. (English) Zbl 0793.58010
The incompressible surfaces play an important role in the deformation theory of hyperbolic 3-manifolds. Such a surface of genus $$g > 1$$, embedded in the manifold $$N$$, may have induced metrics which determine points in the Teichmüller space $${\mathcal T}(S)$$ of conformal (or hyperbolic) structures on $$S$$. It has been conjectured that the locus of these points is related in an appropriate way to a geodesic in $${\mathcal T}(S)$$, and this is true for some known examples [J. Cannon and W. Thurston, ‘Group invariant Peano curves’, preprint (1989); the author, ‘On rigidity, limit sets, and end invariants of hyperbolic 3- manifolds, preprint)].
For any metric $$\sigma$$ on the surface $$S$$ in the hyperbolic manifold $$N$$, one can consider a map $$f_ \sigma: S \to N$$ of least “energy” $${\mathcal E}(f_ \sigma) = {\textstyle{1\over 2}} \int_ N | df_ \sigma |^ 2 dv(N)$$ in the homotopy class below, and ask about the locus of points $$[\sigma]$$ in $${\mathcal T}(S)$$ where $$\mathcal E$$ is bounded above by a given constant. In this direction the author proves
Theorem A. Let $$N = H^ 3/\Gamma$$ be a hyperbolic 3-manifold, $$S$$ a closed surface of genus at least 2, and $$[f: S\to N]$$ a $$\pi_ 1$$- injective homotopy class of maps. Suppose a positive constant $$\varepsilon_ 0$$ so that $$\text{inj}_ N(x) \geq \varepsilon_ 0$$ for all $$x\in N$$. Then there is a Teichmüller geodesic segment, ray, or line $$L$$ in the Teichmüller space $${\mathcal T}(S)$$ and constants $$A$$, $$B$$ depending only on $$\chi(S)$$ and $$\varepsilon_ 0$$ such that
1. Every Riemann surface on $$L$$ can be mapped into $$N$$ by a map in $$[f]$$ with energy at most $$A$$.
2. Every pleated surface $$g: S \to N$$ homotopic to $$f$$ determines an induced hyperbolic metric on $$S$$ that lies in a $$B$$-neighbourhood of $$L$$.
Here the positive lower bound on the injectivity radius is restrictive but crucial. The result is to answer affirmatively Thurston’s “ending lamination conjecture” for hyperbolic manifolds, admitting a positive lower bound on injectivity radius.
Other motivations are also indicated in the paper. These theorems are to appear in the forthcoming paper of the author (the preprint mentioned above).

##### MSC:
 58E11 Critical metrics 32G05 Deformations of complex structures 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 58E20 Harmonic maps, etc. 58E40 Variational aspects of group actions in infinite-dimensional spaces
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