Xuan Hien Nguyen — Page 2

arXiv2007

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

We study the Dirichlet problem associated to the equation for self-similar surfaces for graphs over the Euclidean plane with a disk removed. We show the existence of a solution provided the boundary conditions on the boundary circle are small enough and satisfy some symmetries. This is the second step towards the construction of new examples of complete embedded self similar surfaces under mean curvature flow.

math.DG↗

arXiv2006

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

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.

math.DG↗