Area-contracting maps between rectangles

In this thesis, I worked on estimating the smallest k-dilation of all diffeomorphisms between two n-dimensional rectangles R and S. I proved that for many rectangles there are highly non-linear diffeomorphisms with much smaller k-dilation than any linear diffeomorphism. When k is equal to n-l, I determined the smallest k-dilation up to a constant factor. For all values of k and n, I solved the following related problem up to a constant factor. Given n-dimensional rectangles R and S, decide if there is an embedding of S into R which maps each k-dimensional submanifold of S to an image with larger k-volume. I also applied the k-dilation techniques to two purely topological problems: estimating the Hopf invariant of a map from a 3-manifold to a high-genus surface, and determining whether there is a map of non-zero degree from a 3-manifold to a hyperbolic 3-manifold.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2005

Author dc:creator

• Guth, Lawrence

Advisor dc:contributor.advisor

• Tomasz S. Mrowka.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/31158

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/31158