Ancient asymptotically cylindrical flows and applications

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of mean curvature flow in $${\mathbb {R}}^{n+1}$$ all $$n\ge 3$$ : show that if a $$\{M_t\}$$ has an $$S^{n-1}\times {\mathbb {R}}$$ singularity at $$(x_0,t_0)$$ , then there exists $$\varepsilon =\varepsilon (x_0,t_0)>0$$ such $$M_t\cap B(x_0,\varepsilon )$$ is $$t\in (t_0-\varepsilon ^2,t_0+\varepsilon ^2)$$ . As case $$n=2$$ which was resolved by first three authors Choi et al. (Acta Math, 2018), existence follows from classifying certain class ancient Brakke flows arise as potential blowup limits near singularity. Specifically, any unit-regular integral with cylindrical blowdown must be either round shrinking cylinder, translating bowl soliton, or oval. particular, combined prior result last two (Hershkovits and White Commun Pure Appl Math 73(3):558–580, 2020), obtain uniqueness through singularities. The main difficulty addressing higher dimensional promoting spectral analysis on cylinder to global geometric properties solution. Most crucially, due wide variety self-shrinking entropy lower than when smoothness does not follow soft arguments. This precludes use classical moving plane method derive symmetry. To overcome this, introduce novel variant method, call “moving without assuming smoothness”—where symmetry are established tandem.