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

## TL;DR

This paper proves the existence of certain self-similar surfaces under mean curvature flow, advancing the construction of new complete embedded solutions with specific boundary conditions and symmetries.

## Contribution

It establishes the existence of solutions to the Dirichlet problem for self-similar surfaces over punctured planes, contributing to the construction of novel embedded self-similar surfaces.

## Key findings

- Existence of solutions under small boundary conditions
- Solutions satisfy prescribed symmetries
- Progress towards new embedded self-similar surfaces

## Abstract

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.