In this thesis, we study the complex geometry of the Teichm\u00fcller space of conformal structures on a finite-type Riemann surface. We give partial answers to two structural questions: (1) Which holomorphic disks in Teichm\u00fcller space are holomorphic retracts of Teichm\u00fcller space? (2) What are the holomorphic and Kobayashi-isometric submersions between Teichm\u00fcller 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.
\r\n\r\nQuestion (1) is equivalent to asking the following: on which pairs of points in Teichm\u00fcller space do the Carath\u00e9odory and Teichm\u00fcller metrics coincide? Markovic showed that the Carath\u00e9odory and Teichm\u00fcller metrics on Teichm\u00fcller space are not the same. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichm\u00fcller disk generated by a differential with no odd-order zeros. We conjecture the converse: the Carath\u00e9odory and Teichm\u00fcller metrics agree on a Teichm\u00fcller disk if and only if the Teichm\u00fcller disk is generated by a differential with no odd-order zeros. We prove this conjecture for the Teichm\u00fcller 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(\u211d).
\r\n\r\nQuestion (2) concerns holomorphic and isometric submersions between Teichm\u00fcller spaces of finite-type surfaces. We prove that, with potential exceptions coming from low-genusphenomena, any such map is a forgetful map \u03c4g,n \u2192 \u03c4g,m obtained by filling in punctures. This generalizes a classical result of Royden and Earle-Kra asserting that biholomorphisms between finite-type Teichm\u00fcller spaces arise from mapping classes. As a key step in the argument, we prove that any \u2102-linear embedding Q(X) \u21aa 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 Teichm\u00fcller spaces. The main analytic tool used is a theorem of Rudin on isometries of Lp spaces.
", "doi": "10.7907/XKMM-8591", "publication_date": "2019", "thesis_type": "phd", "thesis_year": "2019" }, { "id": "thesis:11515", "collection": "thesis", "collection_id": "11515", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05132019-171340673", "primary_object_url": { "basename": "GekhtmanDmitriThesis2019Final.pdf", "content": "final", "filesize": 450965, "license": "other", "mime_type": "application/pdf", "url": "/11515/1/GekhtmanDmitriThesis2019Final.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Two Holomorphic Extremal Problems in Teichm\u00fcller Theory", "author": [ { "family_name": "Gekhtman", "given_name": "Dmitri", "orcid": "0000-0002-6473-1115", "clpid": "Gekhtman-Dmitri" } ], "thesis_advisor": [ { "family_name": "Markovic", "given_name": "Vladimir", "clpid": "Markovic-V" } ], "thesis_committee": [ { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" }, { "family_name": "Markovic", "given_name": "Vladimir", "clpid": "Markovic-V" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Ivrii", "given_name": "Oleg", "orcid": "0000-0002-8939-3108", "clpid": "Ivrii-O" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this thesis, we study the complex geometry of the Teichm\u00fcller space of conformal structures on a finite-type Riemann surface. We give partial answers to two structural questions: (1) Which holomorphic disks in Teichm\u00fcller space are holomorphic retracts of Teichm\u00fcller space? (2) What are the holomorphic and Kobayashi-isometric submersions between Teichm\u00fcller 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.
\r\n\r\nQuestion (1) is equivalent to asking the following: on which pairs of points in Teichm\u00fcller space do the Carath\u00e9odory and Teichm\u00fcller metrics coincide? Markovic showed that the Carath\u00e9odory and Teichm\u00fcller metrics on Teichm\u00fcller space are not the same. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichm\u00fcller disk generated by a differential with no odd-order zeros. We conjecture the converse: the Carath\u00e9odory and Teichm\u00fcller metrics agree on a Teichm\u00fcller disk if and only if the Teichm\u00fcller disk is generated by a differential with no odd-order zeros. We prove this conjecture for the Teichm\u00fcller 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(\u211d).
\r\n\r\nQuestion (2) concerns holomorphic and isometric submersions between Teichm\u00fcller spaces of finite-type surfaces. We prove that, with potential exceptions coming from low-genusphenomena, any such map is a forgetful map \u03c4g,n \u2192 \u03c4g,m obtained by filling in punctures. This generalizes a classical result of Royden and Earle-Kra asserting that biholomorphisms between finite-type Teichm\u00fcller spaces arise from mapping classes. As a key step in the argument, we prove that any \u2102-linear embedding Q(X) \u21aa 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 Teichm\u00fcller spaces. The main analytic tool used is a theorem of Rudin on isometries of Lp spaces.
", "doi": "10.7907/XKMM-8591", "publication_date": "2019", "thesis_type": "phd", "thesis_year": "2019" } ]