AI 中文总结
本文证明平均曲率流中非退化颈缩奇点在有限时间内形成，给出其精确渐近轮廓，并证明其在C^2扰动下结构稳定，结合密度定理表明该现象在三维中普遍且稳定。
AI 中文摘要
本文研究了平均曲率流中颈缩奇点的形成、精确渐近行为及结构稳定性。受Colding、Ilmanen和Minicozzi建立的柱状自收缩子静态刚性的启发，我们首先证明了一个动力学刚性结果：在足够大的尺度上，初始图形接近广义柱面且具有局部二次向上弯曲的超曲面，其平均曲率流必然在有限时间内产生颈缩奇点。我们还为这些局部演化图形的轮廓函数建立了尖锐的渐近展开式。我们证明了重标度后的图形半径收敛到具有二次弯曲的特定多项式轮廓，从而证明所产生的奇点是非退化的。最后，我们建立了一个开性定理，表明非退化颈缩在初始数据的C^2扰动下是结构稳定的。结合Szekelyhidi关于三维平均曲率流的最新密度定理，我们的结果证实非退化颈缩构成三维平均曲率流中一种普遍且稳定的现象。
英文摘要
In this paper, we study the formation, precise asymptotics, and structural stability of neckpinch singularities in mean curvature flow. Motivated by the static rigidity of cylindrical self-shrinkers established by Colding, Ilmanen and Minicozzi, we first prove a dynamical rigidity result: mean curvature flow of hypersurfaces that are initially graphically close to a generalized cylinder on a sufficiently large scale, subject to a localized quadratic upward bending, inevitably develop a neckpinch singularity in finite time. We also establish sharp asymptotic expansions for the profile functions of these locally evolving graphs. We show that the rescaled graphical radius converges to a specific polynomial profile with quadratic bending, proving that the resulting singularities are nondegenerate. Finally, we establish an openness theorem showing that nondegenerate neckpinches are structurally stable under C^2 perturbations of the initial data. Combined with recent density theorems for the 3-dimensional mean curvature flow by Szekelyhidi, our results confirm that nondegenerate neckpinches constitute a generic and stable phenomenon in 3-dimensional mean curvature flow.
