Self-similar solutions to the mean curvature flow in Euclidean and Minkowski space

In the first part of this thesis, we give a classification of all self-similar solutions to the curve shortening flow in the Euclidean plane R² and discuss basic properties of the curves. The problem of finding the curves is reduced to the study of a twodimensional system of ODEs with two parameters that determine the type of the self-similar motion. In the second part, we describe all possible self-similar motions of immersed hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding hypersurface equations. Then we present a new two-parameter family of immersed helicoidal surfaces that rotate/translate with constant velocity under the flow. We look at their limiting behaviour as the pitch of the helicoidal motion goes to 0 and compare it with the limiting behaviour of the classical helicoidal minimal surfaces. Finally, we give a classification of the immersed cylinders in the family of constant mean curvature helicoidal surfaces. In the third part, we introduce the mean curvature flow of curves in the Minkowski plane R¹,¹ and give a classification of all the self-similar solutions. In addition, we demonstrate five non-self-similar exact solutions to the flow.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2013

Author dc:creator

• Halldórsson, Höskuldur Pétur

Advisor dc:contributor.advisor

• Tobias H. Colding.

• 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/1721.1/83693

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/83693