Convexity and curvature in Lorentzian geometry

Karr, William Alexander

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https://hdl.handle.net/2142/98106 Copy

Description

Title

Convexity and curvature in Lorentzian geometry

Author(s)

Karr, William Alexander

Issue Date

2017-06-06

Director of Research (if dissertation) or Advisor (if thesis)

Alexander, Stephanie B.

Doctoral Committee Chair(s)

Tyson, Jeremy

Committee Member(s)

• Bishop, Richard L.
• Leininger, Christopher

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Space-time
• Curvature
• Convexity
• Convex functions
• Geodesics

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.

Graduation Semester

2017-08

Type of Resource

text

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http://hdl.handle.net/2142/98106

Copyright and License Information

Copyright 2017 William A. Karr

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