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

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

2011-06-26

arxiv.org/abs/1106.5272v2

Mathematics

Differential Geometry

22 pages, 1 figure, removed the extra bending parameter \varphi, added Definition 21 and Lemma 22 so that the subset \Xi is co

Scientific paper

We present new examples of complete embedded self-similar surfaces under mean curvature by gluing a sphere and a plane. These surfaces have finite genus and are the first examples of self-shrinkers in $\R^3$ that are not rotationally symmetric. The strategy for the construction is to start with a family of initial surfaces by desingularizing the intersection of a sphere and a plane, then solve a perturbation problem to obtain a one parameter family of self-similar surfaces. Although we start with surfaces asymptotic to a plane at infinity, the constructed self-similar surfaces are asymptotic to cones at infinity.

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