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 [\textit{J. Cannon} and \textit{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).