Singularity Structure in Mean Curvature Flow of Mean Convex Sets

In this note we announce results on the mean curvature flow of mean convex sets in 3-dimensions. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components which collapse to asymptotically round spheres. In this note we announce results on the mean curvature flow of mean convex sets; all the statements below have natural generalizations to the setting of Riemannian 3-manifolds, but for the sake of simplicity we will primarily discuss subsets of R here. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components which collapse to asymptotically round spheres. Recall that a one-parameter family of smooth hypersurfaces {Mt} ⊂ R flows by mean curvature if (1) zt = H(z) = ∆Mtz , where z = (z1, . . . , zn+1) are coordinates on R n+1 and H = −Hn is the mean curvature vector. The papers [ES91] and [CGG91] defined a level set flow for any closed subset K of R. This is a 1-parameter family of closed sets Kt ⊂ R with K0 = K (when K is a domain bounded by a smooth compact hypersurface then the evolution of ∂K for a short time interval coincides with the classical mean curvature evolution). Following [Whi00], we say that a compact subset K ⊂ R is mean convex if Kt ⊂ Int(K) for all t > 0. In this case there is also an associated Brakke flow M : t 7→ Mt of rectifiable varifolds [Bra78, Ilm94, Whi00], and the pair (M,K), where K := ⋃ t≥0 Kt × {t} ⊂ R × R is called a mean-convex flow, [Whi03]. The fundamental papers [Whi00, Whi03] developed a far-reaching partial regularity theory for mean curvature flow of mean convex subsets of R. Our results build on [Whi00, Whi03], giving finer understanding of the singularities in the 3-dimensional case. Recall that the main result of [Whi00] asserts that the space time singular set of the region swept out by a mean–convex set in R has parabolic Hausdorff dimension at most (n− 1), and [Whi03] proved Date: October 15, 2003. THC was supported by NSF grant DMS 0104453. BK was supported by NSF grant DMS-0204506.