Abstract
Given a double cone $${\mathcal {C}}$$ with entropy at most two that is symmetric across some hyperplane, we show that any integral Brakke flow coming out of the cone must inherit the reflection symmetry for all time, provided the flow is smooth for a short time. As a corollary we prove that any such flow coming out of a rotationally symmetric double cone must stay rotationally symmetric for all time. We also show the existence of a non-self-similar flow coming out of a double cone with entropy at most two, and give an example of such a flow with a finite time singularity. Additionally, we show the existence of self-expanders with triple junctions, which are exceptions to our main theorem.
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