Abstract/Details

Geodesics of random Riemannian metrics


2010 2010

Other formats: Order a copy

Abstract (summary)

We introduce Riemannian First-Passage Percolation (Riemannian FPP) as a new model of random differential geometry, by considering a random, smooth Riemannian metric on [special characters omitted]. We are motivated in our study by the random geometry of first-passage percolation (FPP), a lattice model which was developed to model fluid flow through porous media. By adapting techniques from standard FPP, we prove a shape theorem for our model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one.

In differential geometry, geodesics are curves which locally minimize length. They need not do so globally: consider great circles on a sphere. For lattice models of FPP, there are many open questions related to minimizing geodesics; similarly, it is interesting from a geometric perspective when geodesics are globally minimizing. In the present study, we show that for any fixed starting direction v, the geodesic starting from the origin in the direction v is not minimizing with probability one. This is a new result which uses the infinitesimal structure of the continuum, and for which there is no equivalent in discrete lattice models of FPP.

Indexing (details)


Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences; First-passage percolation; Motion in a random environment; Riemannian geometry
Title
Geodesics of random Riemannian metrics
Author
LaGatta, Tom
Number of pages
122
Publication year
2010
Degree date
2010
School code
0009
Source
DAI-B 71/07, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
9781124034171
Advisor
Wehr, Jan
Committee member
Glickenstein, David; Kennedy, Thomas G.; Watkins, Joseph C.
University/institution
The University of Arizona
Department
Mathematics
University location
United States -- Arizona
Degree
Ph.D.
Source type
Dissertations & Theses
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
3407962
ProQuest document ID
577638552
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
http://search.proquest.com/docview/577638552/abstract
Access the complete full text

You can get the full text of this document if it is part of your institution's ProQuest subscription.

Try one of the following:

  • Connect to ProQuest through your library network and search for the document from there.
  • Request the document from your library.
  • Go to the ProQuest login page and enter a ProQuest or My Research username / password.