Mean curvature flow from conical singularities

Abstract We prove Ilmanen’s resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. This shows how the mean curvature flow evolves through asymptotically conical singularities. Precisely, we prove that the level set flow of a smooth hypersurface $M^{n}\subset \mathbb{R}^{n+1}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>M</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>⊂</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> , $2\leq n\leq 6$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:mn>6</mml:mn></mml:math> , with an isolated conical singularity is modeled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens.