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Construction of complete embedded self-similar surfaces under mean curvature flow. I. (English) Zbl 1166.53046
The author describes an original way to construct new examples of complete embedded surfaces which evolve self-similarly under the mean curvature flow. The strategy is following: The author takes some pair of known self-similar solutions (planes, spheres, cylinders) and replaces a neighborhood of their intersection with an appropriately bent scaled singly periodic Scherk surface (the core). The resulting non-smooth surface represents an approximate solution. The task is to find a function whose graph over the constructed surface satisfies the equation for self-similar solutions. The author proposes to consider Dirichlet problems for graphs of functions on different pieces of the constructed surface. The first step is carried out: it is shown 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. Dirichlet problems on other pieces as well as the smooth gluing of solutions will be studied in forthcoming papers.

##### MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
##### Keywords:
mean curvature flow; self-similar solution; Scherk surface
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##### References:
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