Weixu Su

On various Teichmüller spaces of a surface of infinite topological type

We show that the length spectrum metric on Teichmuller spaces of surfaces of infinite topological type is complete. We also give related results and examples that compare the length spectrum Teichmuller space with quasiconformal and the Fenchel-Nielsen Teichmuller spaces on such surfaces. AMS Mathematics Subject Classification: 32G15 ; 30F30 ; 30F60.

On the classification of mapping class actions on Thurston's asymmetric metric

We study the action of the elements of the mapping class group of a surface of finite type on the Teichmuller space of that surface equipped with Thurstons asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurstons classification of mapping classes. The study is parallel to the one made by Bers in the setting of Teichmuller space equipped with Teichmullers metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichm¨ space equipped with the Weil-Petersson metric.

The horofunction compactification of the Teichmüller metric

One fundamental theorem in the theory of holomorphic dynamics is Thurstons topological characterization of postcritically finite rational maps. Its proof is a beautiful application of Teichmuller theory. In this chapter we provide a self-contained proof of a slightly generalized version of Thurstons theorem (the marked Thurstons theorem). We also mention some applications and related results, as well as the notion of deformation spaces of rational maps introduced by A. Epstein.1 The spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 The gluing and the operad structures . . . . . . . . . . . . . . . . . . . . . . 11 3 Framed little discs and the Gerstenhaber and BV Structures . . . . . . . . . 23 4 Moduli space, the Sullivan–PROP and (framed) little discs . . . . . . . . . . 35 5 Stops, Stabilization and the Arc spectrum . . . . . . . . . . . . . . . . . . . 40 6 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7 Open/Closed version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 A Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

A family of conformally natural extensions of homeomorphisms of the circle

In this article, we give a family of conformally natural extensions of homeomorphisms of the circle.

Variation of Extremal Length Functions on Teichmüller Space

Extremal length is an important conformal invariant on Riemann surface. It is closely related to the geometry of Teichmuller metric on Teichmuller space. By identifying extremal length functions with energy of harmonic maps from Riemann surfaces to

ON FENCHEL-NIELSEN COORDINATES ON TEICHMÜLLER SPACES OF SURFACES OF INFINITE TYPE

\mathbb{R}

ON LENGTH SPECTRUM METRICS AND WEAK METRICS ON TEICHMÜLLER SPACES OF SURFACES WITH BOUNDARY

-trees, we study the second variation of extremal length functions along Weil-Petersson geodesics. We show that the extremal length of any measured foliation is a pluri-subharmonic function on Teichmuller space.