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Alexander, Stephanie B. Tyson, Jeremy Bishop, Richard L. Leininger, Christopher Karr, William Alexander 2017-09-29T16:37:49Z 2017-09-29T16:37:49Z 2017-06-06 2017-08 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). No. of bitstreams: 6 KARR-DISSERTATION-2017.pdf: 3348732 bytes, checksum: 96954a03bfa34c71154cd9412472f11d (MD5) deSitter.pdf: 281377 bytes, checksum: a7742cc275e35e480102157938ab8a9c (MD5) karr-thesis-16may.tex: 192073 bytes, checksum: a057ffebd4476ceea2af03ffc5031cb5 (MD5) orth1.pdf: 1206133 bytes, checksum: af7c01d3df5ea8ac550ad6816bf88e16 (MD5) orth2.pdf: 1195614 bytes, checksum: 23de3017592a3a92e95a868bad3bd4cf (MD5) LICENSE.txt: 4209 bytes, checksum: 64b74a124e5a0c3b6f6058427bbb2894 (MD5) Previous issue date: 2017-06-06 application/pdf http://hdl.handle.net/2142/98106 en Copyright 2017 William A. Karr Space-time Curvature Convexity Convex functions Geodesics Convexity and curvature in Lorentzian geometry text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.