## Area-contracting maps between rectangles

##### Author(s)

Guth, Lawrence
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##### Other Contributors

Massachusetts Institute of Technology. Dept. of Mathematics.

##### Advisor

Tomasz S. Mrowka.

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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.

##### Description

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. Includes bibliographical references (p. 207-208).

##### Date issued

2005##### Department

Massachusetts Institute of Technology. Dept. of Mathematics.; Massachusetts Institute of Technology. Department of Mathematics##### Publisher

Massachusetts Institute of Technology

##### Keywords

Mathematics.