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Title:  Convexity and curvature in Lorentzian geometry 
Author(s):  Karr, William Alexander 
Director of Research:  Alexander, Stephanie B. 
Doctoral Committee Chair(s):  Tyson, Jeremy 
Doctoral Committee Member(s):  Bishop, Richard L.; Leininger, Christopher 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Spacetime
Curvature Convexity Convex functions Geodesics 
Abstract:  A spacetime 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 spacetime 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 spacetimes. Chapter 3 explores the relation between convex functions and geodesic connectedness of spacetimes. We give geometrictopological proofs of geodesic connectedness for classes of spacetimes to which known methods do not apply. For instance, a nulldisprisoning spacetime 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 spacetimes satisfying $\mathcal{R} \geq 0$) are geodesically connected. Chapter 4 explores the relationship between socalled $\lambda$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 spacetimes 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. 
Issue Date:  20170606 
Type:  Text 
URI:  http://hdl.handle.net/2142/98106 
Rights Information:  Copyright 2017 William A. Karr 
Date Available in IDEALS:  20170929 
Date Deposited:  201708 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois