Evolution of spoon-shaped networks

TL;DR

This paper studies the curvature-driven evolution of a specific spoon-shaped network in a convex domain, showing how the initial enclosed area influences the maximal existence time and describing the asymptotic shape as a Brakke spoon.

Contribution

It provides new insights into the evolution and asymptotic behavior of spoon-shaped networks under curvature flow, linking initial enclosed area to maximal existence time.

Findings

Maximal existence time depends only on initial enclosed area.

The closed curve shrinks to a point during evolution.

The network asymptotically approaches a Brakke spoon shape.

Abstract

We consider a regular embedded network composed by two curves, one of them closed, in a convex domain $\Omega$. The two curves meet only in one point, forming angle of $120$ degrees. The non-closed curve has a fixed end point on $\partial\Omega$. We study the evolution by curvature of this network. We show that the maximal existence time depends only on the area enclosed in the initial loop, if the length of the non-closed curve stays bounded from below during the evolution. Moreover, the closed curve shrinks to a point and the network is asymptotically approaching, after dilations and extraction of a subsequence, a Brakke spoon.