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