{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/322407"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/322407","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Asymptotically cylindrical Calabi–Yau and special Lagrangian geometry","abstract":"We study asymptotically cylindrical Calabi–Yau manifolds and their asymptotically cylindrical special Lagrangian submanifolds. As a prototype problem, we also consider an extension of Hodge theory to general asymptotically cylindrical manifolds. For our study of asymptotically cylindrical Calabi–Yau manifolds, we restrict to complex dimension three. We regard a Calabi–Yau structure as a pair of closed forms ($\\Omega$, $\\omega$); the assumption that the structure is asymptotically cylindrical gives an asymptotic condition on ($\\Omega$, $\\omega$). Regarding the Riemannian products of Calabi-Yau threefolds with S$^{1}$ as G$_{2}$ manifolds, we show that the asymptotically cylindrical deformations of a Calabi–Yau structure (with possibly varying asymptotic limit) are unobstructed. Locally, the spaces of deformations are given by appropriate spaces of harmonic forms. We then show that we can glue asymptotically cylindrical Calabi–Yau manifolds, and that if we do so the “gluing map” of moduli spaces is essentially a local diffeomorphism. In particular, it is an open mapping. In the case of asymptotically cylindrical special Lagrangian submanifolds, we no longer explicitly restrict to dimension three; we assume only that we have a gluing theorem for Calabi–Yau manifolds of the kind obtained in dimension three. McLean and others have constructed deformation spaces of special Lagrangian submanifolds; we show that gluing of asymptotically cylindrical special Lagrangian submanifolds is again unobstructed. As in the Calabi–Yau case, we can define a “gluing map” and this map is a local diffeomorphism of moduli spaces. In both cases, the local diffeomorphism property gives a “local Mayer–Vietoris principle” for deformations. In the special Lagrangian case, the linearisation of the “ungluing” map so defined is just the map of harmonic forms induced by Hodge theory from the natural map of de Rham cohomology; in the Calabi–Yau case it is only slightly more involved.","abstract_html":"We study asymptotically cylindrical Calabi–Yau manifolds and their asymptotically cylindrical special Lagrangian submanifolds. As a prototype problem, we also consider an extension of Hodge theory to general asymptotically cylindrical manifolds. For our study of asymptotically cylindrical Calabi–Yau manifolds, we restrict to complex dimension three. We regard a Calabi–Yau structure as a pair of closed forms ($\\Omega$, <span class=\"etd-inline-math\">&omega;</span>); the assumption that the structure is asymptotically cylindrical gives an asymptotic condition on ($\\Omega$, <span class=\"etd-inline-math\">&omega;</span>). Regarding the Riemannian products of Calabi-Yau threefolds with S<span class=\"etd-inline-math\"><sup>1</sup></span> as G<span class=\"etd-inline-math\"><sub>2</sub></span> manifolds, we show that the asymptotically cylindrical deformations of a Calabi–Yau structure (with possibly varying asymptotic limit) are unobstructed. Locally, the spaces of deformations are given by appropriate spaces of harmonic forms. We then show that we can glue asymptotically cylindrical Calabi–Yau manifolds, and that if we do so the “gluing map” of moduli spaces is essentially a local diffeomorphism. In particular, it is an open mapping. In the case of asymptotically cylindrical special Lagrangian submanifolds, we no longer explicitly restrict to dimension three; we assume only that we have a gluing theorem for Calabi–Yau manifolds of the kind obtained in dimension three. McLean and others have constructed deformation spaces of special Lagrangian submanifolds; we show that gluing of asymptotically cylindrical special Lagrangian submanifolds is again unobstructed. As in the Calabi–Yau case, we can define a “gluing map” and this map is a local diffeomorphism of moduli spaces. In both cases, the local diffeomorphism property gives a “local Mayer–Vietoris principle” for deformations. In the special Lagrangian case, the linearisation of the “ungluing” map so defined is just the map of harmonic forms induced by Hodge theory from the natural map of de Rham cohomology; in the Calabi–Yau case it is only slightly more involved.","abstract_has_math":true,"creators":["Talbot, Timothy James"],"institution":"University of Cambridge","degree_name":"PhD","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Kovalev, Alexei"],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-04-13","date_published":"2017-04-13","updated_at":"2026-07-22T22:24:00Z","subjects":["Asymptotically cylindrical Calabi–Yau manifolds","Asymptotically cylindrical special Lagrangian submanifolds"],"languages":["en"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/c16610e2-fd40-4554-9243-ebcabf09218e/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.69864","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Kovalev, Alexei"]},{"key":"dc:creator","label":"Author","values":["Talbot, Timothy James"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2017-04-13"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/322407"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["PhD"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Asymptotically cylindrical Calabi–Yau manifolds","Asymptotically cylindrical special Lagrangian submanifolds"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/c16610e2-fd40-4554-9243-ebcabf09218e/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.69864"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/951ee218-89fe-4dd6-843e-1a5deaf6c4ed/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We study asymptotically cylindrical Calabi–Yau manifolds and their asymptotically cylindrical special Lagrangian submanifolds. As a prototype problem, we also consider an extension of Hodge theory to general asymptotically cylindrical manifolds. For our study of asymptotically cylindrical Calabi–Yau manifolds, we restrict to complex dimension three. We regard a Calabi–Yau structure as a pair of closed forms ($\\Omega$, $\\omega$); the assumption that the structure is asymptotically cylindrical gives an asymptotic condition on ($\\Omega$, $\\omega$). Regarding the Riemannian products of Calabi-Yau threefolds with S$^{1}$ as G$_{2}$ manifolds, we show that the asymptotically cylindrical deformations of a Calabi–Yau structure (with possibly varying asymptotic limit) are unobstructed. Locally, the spaces of deformations are given by appropriate spaces of harmonic forms. We then show that we can glue asymptotically cylindrical Calabi–Yau manifolds, and that if we do so the “gluing map” of moduli spaces is essentially a local diffeomorphism. In particular, it is an open mapping. In the case of asymptotically cylindrical special Lagrangian submanifolds, we no longer explicitly restrict to dimension three; we assume only that we have a gluing theorem for Calabi–Yau manifolds of the kind obtained in dimension three. McLean and others have constructed deformation spaces of special Lagrangian submanifolds; we show that gluing of asymptotically cylindrical special Lagrangian submanifolds is again unobstructed. As in the Calabi–Yau case, we can define a “gluing map” and this map is a local diffeomorphism of moduli spaces. In both cases, the local diffeomorphism property gives a “local Mayer–Vietoris principle” for deformations. In the special Lagrangian case, the linearisation of the “ungluing” map so defined is just the map of harmonic forms induced by Hodge theory from the natural map of de Rham cohomology; in the Calabi–Yau case it is only slightly more involved."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["19b2914b29a88bebf6e601307d2b8f32","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["Asymptotically cylindrical Calabi–Yau and special Lagrangian geometry"]}]}],"canonical_facts":{"dc:contributor.advisor":["Kovalev, Alexei"],"dc:creator":["Talbot, Timothy James"],"dc:date.issued":["2017-04-13"],"dc:description.abstract":["We study asymptotically cylindrical Calabi–Yau manifolds and their asymptotically cylindrical special Lagrangian submanifolds. As a prototype problem, we also consider an extension of Hodge theory to general asymptotically cylindrical manifolds. For our study of asymptotically cylindrical Calabi–Yau manifolds, we restrict to complex dimension three. We regard a Calabi–Yau structure as a pair of closed forms ($\\Omega$, $\\omega$); the assumption that the structure is asymptotically cylindrical gives an asymptotic condition on ($\\Omega$, $\\omega$). Regarding the Riemannian products of Calabi-Yau threefolds with S$^{1}$ as G$_{2}$ manifolds, we show that the asymptotically cylindrical deformations of a Calabi–Yau structure (with possibly varying asymptotic limit) are unobstructed. Locally, the spaces of deformations are given by appropriate spaces of harmonic forms. We then show that we can glue asymptotically cylindrical Calabi–Yau manifolds, and that if we do so the “gluing map” of moduli spaces is essentially a local diffeomorphism. In particular, it is an open mapping. In the case of asymptotically cylindrical special Lagrangian submanifolds, we no longer explicitly restrict to dimension three; we assume only that we have a gluing theorem for Calabi–Yau manifolds of the kind obtained in dimension three. McLean and others have constructed deformation spaces of special Lagrangian submanifolds; we show that gluing of asymptotically cylindrical special Lagrangian submanifolds is again unobstructed. As in the Calabi–Yau case, we can define a “gluing map” and this map is a local diffeomorphism of moduli spaces. In both cases, the local diffeomorphism property gives a “local Mayer–Vietoris principle” for deformations. In the special Lagrangian case, the linearisation of the “ungluing” map so defined is just the map of harmonic forms induced by Hodge theory from the natural map of de Rham cohomology; in the Calabi–Yau case it is only slightly more involved."],"dc:format.checksum.md5":["19b2914b29a88bebf6e601307d2b8f32","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["10.17863/CAM.69864"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/951ee218-89fe-4dd6-843e-1a5deaf6c4ed/download"],"dc:language":["en"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/322407"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/c16610e2-fd40-4554-9243-ebcabf09218e/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["Asymptotically cylindrical Calabi–Yau manifolds","Asymptotically cylindrical special Lagrangian submanifolds"],"dc:title":["Asymptotically cylindrical Calabi–Yau and special Lagrangian geometry"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["PhD"]},"updated_at":"2026-07-22T22:24:00Z"}