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

*(English)*Zbl 1166.53046The 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.

Reviewer: Vasyl Gorkaviy (Kharkov)

##### MSC:

53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |

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\textit{X. H. Nguyen}, Trans. Am. Math. Soc. 361, No. 4, 1683--1701 (2009; Zbl 1166.53046)

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##### References:

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