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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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: http://dx.doi.org/10.1090/S0002-9947-08-04748-X
PII: S 0002-9947(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.