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We prove the Multiplicity One Conjecture for mean curvature flows of surfacesin $\mathbb{R}^3$. Specifically, we show that any blow-up limit of such meancurvature flows has multiplicity one. This has several applications. First,combining our work with results of Brendle andChoi-Haslhofer-Hershkovits-White, we show that any level set flow starting froman embedded surface diffeomorphic to a 2-spheres does not fatten. In fact, weobtain that the problem of evolving embedded 2-spheres via the mean curvatureflow equation is well-posed within a natural class of singular solutions.Second, we use our result to remove an additional condition in recent work ofChodosh-Choi-Mantoulidis-Schulze. This shows that mean curvature flows startingfrom any generic embedded surface only incur cylindrical or sphericalsingularities. Third, our approach offers a new regularity theory for solutionsof mean curvature flows that flow through singularities. Among other things,this theory also applies to the innermost and outermost flow of any embeddedsurface and shows that all singularity models of such flows must havemultiplicity one. It also establishes equality of the fattening time with thediscrepancy time. Lastly, we obtain a number of further results characterizinga separation phenomenon of mean curvature flows of surfaces.

On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces