论文标题
奇异最小超曲面I的变形，孤立的奇异性
Deformations of Singular Minimal Hypersurfaces I, Isolated Singularities
论文作者
论文摘要
当地稳定的最小超出表面可能具有尺寸$ \ geq 7 $的奇异性，在欧几里得空间中以稳定的稳定和最小化锥体为模型。在本文中，我们介绍了这些奇异性如何影响最小曲面的局部行为的不同方面。 首先，考虑到非分化的最小超浮雕，在每个单点的奇异点上严格稳定，严格最大程度地减少切线锥，在公制的任何小扰动下，我们表明在新的度量标准下存在附近最小的高度。对于扰动方向的一般选择，我们显示了所得最小超出表面的整个平滑度。 其次，考虑到一个严格稳定的最小超出表面$σ$，在每个单个点上严格最大程度地减少了切线锥，我们表明$σ$在其附近的同源性最小化。 最后，鉴于一个稳定的最小超曲面多重性$ 1 $收敛于八个歧管中的稳定最小的高表面$σ$，我们表明存在于这些融合的低表冲突所引起的$σ$上的$σ$上，这些融合的张力是$σ$，这使Ben Sharp sharp sharp clase clase clase clase clase clastion clastion close;此外，该雅各比场的阳性意味着融合序列的平滑度。
Locally stable minimal hypersurface could have singularities in dimension $\geq 7$ in general, locally modeled on stable and area-minimizing cones in the Euclidean spaces. In this paper, we present different aspects of how these singularities may affect the local behavior of minimal hypersurfaces. First, given a non-degenerate minimal hypersurface with strictly stable and strictly minimizing tangent cone at each singular point, under any small perturbation of the metric, we show the existence of a nearby minimal hypersurface under new metric. For a generic choice of perturbation direction, we show the entire smoothness of the resulting minimal hypersurface. Second, given a strictly stable minimal hypersurface $Σ$ with strictly minimizing tangent cone at each singular point, we show that $Σ$ is uniquely homologically minimizing in its neighborhood. Lastly, given a family of stable minimal hypersurfaces multiplicity $1$ converges to a stable minimal hypersurface $Σ$ in an eight manifold, we show that there exists a non-trivial Jacobi field on $Σ$ induced by these converging hypersurfaces, which generalizes a result by Ben Sharp in smooth case; Moreover, positivity of this Jacobi field implies smoothness of the converging sequence.
