Two Holomorphic Extremal Problems in Teichmüller Theory

Abstract

In this thesis, we study the complex geometry of the Teichmuller space of conformal structures on a finite-type Riemann surface. We give partial answers to two structural questions: (1) Which holomorphic disks in Teichmuller space are holomorphic retracts of Teichmuller space? (2) What are the holomorphic and Kobayashi-isometric submersions between Teichmuller spaces? In both cases, the answers have to do with the geometry of the underlying surfaces, while the methods require developing and applying novel analytic tools. Question (1) is equivalent to asking the following: on which pairs of points in Teichmuller space do the Caratheodory and Teichmuller metrics coincide? Markovic showed that the Caratheodory and Teichmuller metrics on Teichmuller space are not the same. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichmuller disk generated by a differential with no odd-order zeros. We conjecture the converse: the Caratheodory and Teichmuller metrics agree on a Teichmuller disk if and only if the Teichmuller disk is generated by a differential with no odd-order zeros. We prove this conjecture for the Teichmuller spaces of the five-times punctured sphere and the twice-punctured torus. As a key analytic step in the proof, we study the family of holomorphic retractions from the polydisk onto its diagonal. In particular, we analyze the asymptotics of the orbit of such a retraction under the conjugation action of a unipotent subgroup of PSL2(ℝ). Question (2) concerns holomorphic and isometric submersions between Teichmuller spaces of finite-type surfaces. We prove that, with potential exceptions coming from low-genusphenomena, any such map is a forgetful map τg,n → τg,m obtained by filling in punctures. This generalizes a classical result of Royden and Earle-Kra asserting that biholomorphisms between finite-type Teichmuller spaces arise from mapping classes. As a key step in the argument, we prove that any ℂ-linear embedding Q(X) ↪ Q(Y) between spaces of holomorphic integrable quadratic differentials is, up to scale, pull-back by a holomorphic map. We accomplish this step by adapting methods developed by Markovic to study isometries of infinite-type Teichmuller spaces. The main analytic tool used is a theorem of Rudin on isometries of Lp spaces.