Construction of complete embedded self-similar surfaces under mean curvature flow. Part I.

Author:
Xuan Hien Nguyen

Journal:
Trans. Amer. Math. Soc. **361** (2009), 1683-1701

MSC (2000):
Primary 53C44

DOI:
https://doi.org/10.1090/S0002-9947-08-04748-X

Published electronically:
November 25, 2008

MathSciNet review:
2465812

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Abstract: We carry out the first main step towards the construction of new examples of complete embedded self-similar surfaces under mean curvature flow. An approximate solution is obtained by taking two known examples of self-similar surfaces and desingularizing the intersection circle using an appropriately modified singly periodic Scherk surface, called the core. Using an inverse function theorem, we show that for small boundary conditions on the core, there is an embedded surface close to the core that is a solution of the equation for self-similar surfaces. This provides us with an adequate central piece to substitute for the intersection.

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Additional Information

**Xuan Hien Nguyen**

Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Address at time of publication:
Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, Kansas 66506

DOI:
https://doi.org/10.1090/S0002-9947-08-04748-X

Keywords:
Mean curvature flow,
self-similar,
singularities

Received by editor(s):
September 13, 2006

Published electronically:
November 25, 2008

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.