Construction of Complete Embedded Self-Similar Surfaces under Mean Curvature Flow. Part I

Details Construction of Complete Embedded Self-Similar Surfaces under Mean Curvature Flow. Part I Construction of Complete Embedded Self-Similar Surfaces under Mean Curvature Flow. Part I

2006-10-23

arxiv.org/abs/math/0610695v1

Trans. of the Amer. Math. Soc. 361 (2009), 1683-1701

Mathematics

Differential Geometry

24 pages, 3 pictures

Scientific paper

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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