Non-Existence of Theta-Shaped Self-Similarly Shrinking Networks Moving by Curvature

TL;DR

This paper proves that theta-shaped networks with two triple junctions cannot shrink self-similarly under curvature flow, completing the classification of such networks with limited junctions.

Contribution

It establishes the non-existence of theta-shaped self-similarly shrinking networks with two triple junctions, filling a gap in the classification of planar curvature flows.

Findings

No theta-shaped networks shrink self-similarly under curvature.

Completes classification of networks with up to two triple junctions.

Advances understanding of curvature-driven network evolution.

Abstract

We prove that there are no networks homeomorphic to the Greek "theta" letter (a double cell) embedded in the plane with two triple junctions with angles of $120$ degrees, such that under the motion by curvature they are self-similarly shrinking. This fact completes the classification of the self-similarly shrinking networks in the plane with at most two triple junctions.