Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness

The equation of course also makes sense in higher dimension and co-dimension and in other ambient manifolds.In this paper, however, we focus on evolving surfaces in R 3 .k. choi, r. haslhofer and o. hershkovits Classification of ancient low-entropy flowsLet M ⊂R 3 be a surface.The entropy, introduced by Colding-Minicozzi [CM1], is defined as the supremum of the Gaussian area over all centers and all scales, namely Ent[M ] = sup(1.2)The entropy measures, in a certain sense, the complexity of the surface.For example, the values for a plane, sphere and cylinder are Ent[R 2 ] = 1, Ent[S 2 ] = 4 e ∼ 1.47, and Ent[S 1 ×R] = 2π e ∼ 1.52.If M={M t } t∈I evolves by mean-curvature flow, then t !Ent[M t ] is non-increasing by Husiken's monotonicity formula [Hu2], and hence Ent[M] := sup t∈I Ent[M t ] = lim t!inf(I) Ent[M t ].For example, the entropy of a flat plane P, a round shrinking sphere S, a round shrinking cylinder Z, a translating bowl soliton B, and an ancient oval O, are given by Ent[P] = 1, Ent[S] = 4 e ∼ 1.47 Ent[Z] = Ent[B] = Ent[O] = 2π e ∼ 1.52.By a beautiful classification result of Bernstein-Wang [BW] any self-similarly shrinking mean-curvature flow in R 3 with entropy at most 2π/e is either a flat plane, round shrinking sphere, or round shrinking cylinder.By a recent uniqueness result of Hershkovits [He] any self-similarly translating flow in R 3 with entropy at most 2π/e is a bowl soliton.These results in turn build on a pioneering paper by Colding-Ilmanen-Minicozzi-White [CIMW] that was the genesis of the study of low-entropy flows.For the study of singularities, it is important that these concepts are also available in the non-smooth setting.Singular surfaces in Euclidean space are described most easily by 2-rectifiable Radon measures, which generalize the area measure of smooth 2-dimensional surfaces; see e.g.[Si], [I1].Recall that a 2-rectifiable Radon measure µ in R 3 is a Radon measure that has a 2-dimensional tangent plane of positive multiplicity at almost every point.The entropy of µ is defined as Ent[µ] = sup

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