Uniqueness of collapsed simply connected translating solitons of low entropy

Author: Niels Martin

Abstract: We consider self-translating solitons for the mean curvature flow of complete embedded 2-surfaces of finite genus and finite entropy. Under a collapsedness condition - amounting to the confinement into slabs of 3-space - we first define the concepts of "wing types" and "wing numbers". In terms of these, a simple formula then computes the entropy, which is in particular quantized into integer values. Asking further which examples of solitons exist in the low entropy range, we combine the wing number ideas with Morse theory for minimal surfaces a la Radó to prove a uniqueness theorem: The unique simply connected complete embedded translating solitons contained in slabs and of entropy $\lambda(\Sigma) = 3$ are the Hoffman-Martín-White pitchforks. This is joint work with E.S. Gama and F. Martín.