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

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The main goal of the project is the construction of new examples of complete embedded self-similar surfaces under the mean curvature flow, which is the gradient flow of the surface area. The strategy for the construction is to desingularize the intersection of two known examples by adapting the method N. Kapouleas used to construct embedded minimal surfaces; the Principal Investigator will use a Scherk's singly periodic minimal surface to get an approximate solution which is then perturbed to obtain a complete self-similar surface. The project is very geometric in nature, but it involves many techniques from nonlinear partial differential equations. Self-similar surfaces model the behavior of the mean curvature flow near singularities under certain conditions; therefore, the availability of examples of such surfaces is important to the understanding of the flow near its singularities and its continuation past the singularities. Unfortunately, there are currently only four know examples of complete self-similar surfaces in dimension two embedded in the Euclidean space. A general classification is hopeless; therefore, it is essential to find successful methods to construct new examples. The broader goal of this project is to obtain a better understanding of the formation of singularities for the mean curvature flow.