Two Holomorphic Extremal Problems in Teichmüller Theory

Citation

Gekhtman, Dmitri (2019) Two Holomorphic Extremal Problems in Teichmüller Theory. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/XKMM-8591. https://resolver.caltech.edu/CaltechTHESIS:05132019-171340673

Abstract

In this thesis, we study the complex geometry of the Teichmüller space of conformal structures on a finite-type Riemann surface. We give partial answers to two structural questions: (1) Which holomorphic disks in Teichmüller space are holomorphic retracts of Teichmüller space? (2) What are the holomorphic and Kobayashi-isometric submersions between Teichmüller 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 Teichmüller space do the Carathéodory and Teichmüller metrics coincide? Markovic showed that the Carathéodory and Teichmüller metrics on Teichmüller space are not the same. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichmüller disk generated by a differential with no odd-order zeros. We conjecture the converse: the Carathéodory and Teichmüller metrics agree on a Teichmüller disk if and only if the Teichmüller disk is generated by a differential with no odd-order zeros. We prove this conjecture for the Teichmüller 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 Teichmüller spaces of finite-type surfaces. We prove that, with potential exceptions coming from low-genusphenomena, any such map is a forgetful map τ_g,n → τ_g,m obtained by filling in punctures. This generalizes a classical result of Royden and Earle-Kra asserting that biholomorphisms between finite-type Teichmüller 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 Teichmüller spaces. The main analytic tool used is a theorem of Rudin on isometries of L^p spaces.

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