{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/98106"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/98106","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Convexity and curvature in Lorentzian geometry","abstract":"A space-time satisfies $\\mathcal{R} \\geq K $ if the sectional curvatures are bounded below by $K$ for spacelike planes and above by $K$ for timelike planes (similarly, a space-time satisfies $\\mathcal{R} \\leq K$ if the aforementioned inequalities are reversed). We demonstrate that these curvature bound conditions together with convex functions are effective means to study the geometry of space-times. Chapter 3 explores the relation between convex functions and geodesic connectedness of space-times. We give geometric-topological proofs of geodesic connectedness for classes of space-times to which known methods do not apply. For instance, a null-disprisoning space-time is geodesically connected if it supports a proper, nonnegative strictly convex function whose critical set is a point. In particular, timelike strictly convex hypersurfaces of Minkowski space (which are prototypical examples of space-times satisfying $\\mathcal{R} \\geq 0$) are geodesically connected. Chapter 4 explores the relationship between so-called $\\lambda$-convex functions ($ \\hess f(x,x) \\geq \\lambda \\langle x,x \\rangle $), curvature bounds, and trapped submanifolds. We show that certain types of trapped submanifolds can be ruled out for domains of space-times satisfying $\\mathcal{R} \\leq K$. Using the full curvature bound condition $\\mathcal{R} \\leq K$ allows us to extend previous results that use timelike sectional curvature bounds to rule out trapped submanifolds in the chronological future of a point.","abstract_html":"A space-time satisfies $\\mathcal{R} \\geq K $ if the sectional curvatures are bounded below by $K$ for spacelike planes and above by $K$ for timelike planes (similarly, a space-time satisfies $\\mathcal{R} \\leq K$ if the aforementioned inequalities are reversed). We demonstrate that these curvature bound conditions together with convex functions are effective means to study the geometry of space-times. Chapter 3 explores the relation between convex functions and geodesic connectedness of space-times. We give geometric-topological proofs of geodesic connectedness for classes of space-times to which known methods do not apply. For instance, a null-disprisoning space-time is geodesically connected if it supports a proper, nonnegative strictly convex function whose critical set is a point. In particular, timelike strictly convex hypersurfaces of Minkowski space (which are prototypical examples of space-times satisfying $\\mathcal{R} \\geq 0$) are geodesically connected. Chapter 4 explores the relationship between so-called $\\lambda$-convex functions ($ \\hess f(x,x) \\geq \\lambda \\langle x,x \\rangle $), curvature bounds, and trapped submanifolds. We show that certain types of trapped submanifolds can be ruled out for domains of space-times satisfying $\\mathcal{R} \\leq K$. Using the full curvature bound condition $\\mathcal{R} \\leq K$ allows us to extend previous results that use timelike sectional curvature bounds to rule out trapped submanifolds in the chronological future of a point.","abstract_has_math":true,"creators":["Karr, William Alexander"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Alexander, Stephanie B.","Tyson, Jeremy","Bishop, Richard L.","Leininger, Christopher"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-09-29T16:37:49Z","date_published":"2017-09-29T16:37:49Z","updated_at":"2026-07-22T22:24:34Z","subjects":["Space-time","Curvature","Convexity","Convex functions","Geodesics"],"languages":["en"],"rights":["Copyright 2017 William A. 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Karr"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/98106"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A space-time satisfies $\\mathcal{R} \\geq K $ if the sectional curvatures are bounded below by $K$ for spacelike planes and above by $K$ for timelike planes (similarly, a space-time satisfies $\\mathcal{R} \\leq K$ if the aforementioned inequalities are reversed). We demonstrate that these curvature bound conditions together with convex functions are effective means to study the geometry of space-times. Chapter 3 explores the relation between convex functions and geodesic connectedness of space-times. We give geometric-topological proofs of geodesic connectedness for classes of space-times to which known methods do not apply. For instance, a null-disprisoning space-time is geodesically connected if it supports a proper, nonnegative strictly convex function whose critical set is a point. In particular, timelike strictly convex hypersurfaces of Minkowski space (which are prototypical examples of space-times satisfying $\\mathcal{R} \\geq 0$) are geodesically connected. Chapter 4 explores the relationship between so-called $\\lambda$-convex functions ($ \\hess f(x,x) \\geq \\lambda \\langle x,x \\rangle $), curvature bounds, and trapped submanifolds. We show that certain types of trapped submanifolds can be ruled out for domains of space-times satisfying $\\mathcal{R} \\leq K$. Using the full curvature bound condition $\\mathcal{R} \\leq K$ allows us to extend previous results that use timelike sectional curvature bounds to rule out trapped submanifolds in the chronological future of a point.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms","The student, William Karr, accepted the attached license on 2017-05-30 at 12:23.","The student, William Karr, submitted this Dissertation for approval on 2017-05-30 at 12:36.","This Dissertation was approved for publication on 2017-06-06 at 15:15.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11185 on 2017-09-29 at 11:25:52","Made available in DSpace on 2017-09-29T16:37:49Z (GMT). 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We demonstrate that these curvature bound conditions together with convex functions are effective means to study the geometry of space-times. Chapter 3 explores the relation between convex functions and geodesic connectedness of space-times. We give geometric-topological proofs of geodesic connectedness for classes of space-times to which known methods do not apply. For instance, a null-disprisoning space-time is geodesically connected if it supports a proper, nonnegative strictly convex function whose critical set is a point. In particular, timelike strictly convex hypersurfaces of Minkowski space (which are prototypical examples of space-times satisfying $\\mathcal{R} \\geq 0$) are geodesically connected. Chapter 4 explores the relationship between so-called $\\lambda$-convex functions ($ \\hess f(x,x) \\geq \\lambda \\langle x,x \\rangle $), curvature bounds, and trapped submanifolds. We show that certain types of trapped submanifolds can be ruled out for domains of space-times satisfying $\\mathcal{R} \\leq K$. Using the full curvature bound condition $\\mathcal{R} \\leq K$ allows us to extend previous results that use timelike sectional curvature bounds to rule out trapped submanifolds in the chronological future of a point.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms","The student, William Karr, accepted the attached license on 2017-05-30 at 12:23.","The student, William Karr, submitted this Dissertation for approval on 2017-05-30 at 12:36.","This Dissertation was approved for publication on 2017-06-06 at 15:15.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11185 on 2017-09-29 at 11:25:52","Made available in DSpace on 2017-09-29T16:37:49Z (GMT). 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Karr"],"dc:subject":["Space-time","Curvature","Convexity","Convex functions","Geodesics"],"dc:title":["Convexity and curvature in Lorentzian geometry"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:34Z"}