Skip to main content
Duke University Libraries
DukeSpace Scholarship by Duke Authors
  • Login
  • Ask
  • Menu
  • Login
  • Ask a Librarian
  • Search & Find
  • Using the Library
  • Research Support
  • Course Support
  • Libraries
  • About
View Item 
  •   DukeSpace
  • Theses and Dissertations
  • Duke Dissertations
  • View Item
  •   DukeSpace
  • Theses and Dissertations
  • Duke Dissertations
  • View Item
JavaScript is disabled for your browser. Some features of this site may not work without it.

Uniformly Area Expanding Flows in Spacetimes

Thumbnail
View / Download
1.9 Mb
Date
2014
Author
Xu, Hangjun
Advisor
Bray, Hubert L
Repository Usage Stats
275
views
500
downloads
Abstract

The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist infinitely many spacetimes, not necessarily spherically symmetric or static, that admit smooth global solutions to inverse mean curvature vector flow. Prior to our work, such solutions were only known in spherically symmetric and static spacetimes. The technique used in this thesis might be important to prove the Spacetime Penrose Conjecture, which remains open today.

Given a spacetime $(N^{4}, \gbar)$ and a spacelike hypersurface $M$. For any closed surface $\Sigma$ embedded in $M$ satisfying some natural conditions, one can ``steer'' the spacetime metric $\gbar$ such that the mean curvature vector field of $\Sigma$ becomes tangential to $M$ while keeping the induced metric on $M$. This can be used to construct more examples of smooth solutions to inverse mean curvature vector flow from smooth solutions to inverse mean curvature flow in a spacelike hypersurface.

Type
Dissertation
Department
Mathematics
Subject
Mathematics
General Relativity
Inverse Mean Curvature Vector Flow
Monotonicity of Hawking Mass
Straight Out Flow
Uniformly Area Expanding Flow
Permalink
https://hdl.handle.net/10161/8732
Citation
Xu, Hangjun (2014). Uniformly Area Expanding Flows in Spacetimes. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/8732.
Collections
  • Duke Dissertations
More Info
Show full item record
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License.

Rights for Collection: Duke Dissertations


Works are deposited here by their authors, and represent their research and opinions, not that of Duke University. Some materials and descriptions may include offensive content. More info

Make Your Work Available Here

How to Deposit

Browse

All of DukeSpaceCommunities & CollectionsAuthorsTitlesTypesBy Issue DateDepartmentsAffiliations of Duke Author(s)SubjectsBy Submit DateThis CollectionAuthorsTitlesTypesBy Issue DateDepartmentsAffiliations of Duke Author(s)SubjectsBy Submit Date

My Account

LoginRegister

Statistics

View Usage Statistics
Duke University Libraries

Contact Us

411 Chapel Drive
Durham, NC 27708
(919) 660-5870
Perkins Library Service Desk

Digital Repositories at Duke

  • Report a problem with the repositories
  • About digital repositories at Duke
  • Accessibility Policy
  • Deaccession and DMCA Takedown Policy

TwitterFacebookYouTubeFlickrInstagramBlogs

Sign Up for Our Newsletter
  • Re-use & Attribution / Privacy
  • Harmful Language Statement
  • Support the Libraries
Duke University