Extremal length geometry of Teichmüller space.

*(English)*Zbl 0702.32019Assume \(\tau\) is a point in the Teichmüller space of a Riemann surface which is compact or obtainable from a compact surface by deleting a finite number of punctures. Let \(ext_{\tau}(| du|)\) and \(ext_{\tau}(| dv|)\) be extremal lengths of two transversely realizable measured foliations on a Riemann surface \(R_{\tau}\) corresponding to a point \(\tau\). It is shown that there is a unique Teichmüller line along which the product \(F(\tau)=ext_{\tau}(| du|)ext_{\tau}(| dv|)\) is minimum. In the case that the measured foliations \(| du|\) and \(| dv|\) are transverse measures associated with a pseudo-Anosov diffeomorphism, this line is the unique invariant axis for the element of the mapping class group corresponding to the diffeomorphism.

Teichmüller space embeds into projective classes of vectors of square roots of extremal lengths of simple curves on the base surface. The closure of the image of Teichmüller space under this embedding is compact. Moreover, because we take square roots of extremal lengths, there is a relationship between the boundary of this embedding and the boundary of the Thurston embedding by projective classes of vectors of Poincaré lengths. The boundary of the extremal length embedding property contains the Thurston boundary.

Teichmüller space embeds into projective classes of vectors of square roots of extremal lengths of simple curves on the base surface. The closure of the image of Teichmüller space under this embedding is compact. Moreover, because we take square roots of extremal lengths, there is a relationship between the boundary of this embedding and the boundary of the Thurston embedding by projective classes of vectors of Poincaré lengths. The boundary of the extremal length embedding property contains the Thurston boundary.

Reviewer: F.P.Gardiner

##### MSC:

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

30F60 | Teichmüller theory for Riemann surfaces |

30F45 | Conformal metrics (hyperbolic, Poincaré, distance functions) |

37D99 | Dynamical systems with hyperbolic behavior |