Teichmüller geodesics and ends of hyperbolic 3-manifolds.

*(English)*Zbl 0793.58010The 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).

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).

Reviewer: E.Molnár (Budapest)