2024/12/13 by Kyeongsu Choi, Choi, Kyeongsu, Robert Haslhofer +1
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2412.10581
openalex publication_date 2024/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we classify all noncollapsed singularities of the mean curvature flow in ℝ4. Specifically, we prove that any ancient noncollapsed solution either is one of the classical historical examples (namely ℝj× S3-j, ℝ× 2d-bowl, ℝ× 2d-oval, the rotationally symmetric 3d-bowl, or a cohomogeneity-one 3d-oval), or belongs to the 1-parameter family of ℤ2× O2-symmetric 3d-translators constructed by Hoffman-Ilmanen-Martin-White, or belongs to the 1-parameter family of ℤ22× O2-symmetric ancient 3d-ovals constructed by Du-Haslhofer. In light of the five prior papers on the classification program in ℝ4 from our collaborations with Du, Hershkovits, and Choi-Daskalopoulos-Sesum, the major remaining challenge is the case of mixed behaviour, where the convergence to the round bubble-sheet is fast in x1-direction, but logarithmically slow in x2-direction. To address this, we prove a differential neck theorem, which allows us to capture the (dauntingly small) slope in x1-direction. To establish the differential neck theorem, we introduce a slew of new ideas of independent interest, including switch and differential Merle-Zaag dynamics, anisotropic barriers, and propagation of smallness estimates. Applying our differential neck theorem, we show that every noncompact strictly convex solution is selfsimilarly translating, and also rule out exotic ovals.