论文标题
最小值的最小值的奇异行为和一般规律性
Singular behavior and generic regularity of min-max minimal hypersurfaces
论文作者
论文摘要
我们表明，对于带有正曲率曲率的普通$ 8 $维riemannian歧管，有一个光滑的最小超出表面。如果没有弯曲条件，我们表明，对于一组密集的8维里曼尼亚指标，至多与一个单数点具有最小的高度表面。这扩展了以前关于仅处理面积最小化超曲面的通用规律性的工作。这些结果是对单参数最小值最小超截面$σ\ subset（m，g）$（在任何维度上有效）的结果估计的结果：$ \ Mathcal H^{0}（\ Mathcal {s s} $ \ MATHCAL {S} _ {nm}（σ）$表示$σ$的单数点集，其独特的切线锥体在任何一侧都不限制。
We show that for a generic $8$-dimensional Riemannian manifold with positive Ricci curvature, there exists a smooth minimal hypersurface. Without the curvature condition, we show that for a dense set of 8-dimensional Riemannian metrics there exists a minimal hypersurface with at most one singular point. This extends previous work on generic regularity that only dealt with area-minimizing hypersurfaces. These results are a consequence of a more general estimate for a one-parameter min-max minimal hypersurface $Σ\subset (M,g)$ (valid in any dimension): $$\mathcal H^{0} (\mathcal{S}_{nm}(Σ)) +{\rm Index}(Σ) \leq 1$$ where $\mathcal{S}_{nm}(Σ)$ denotes the set of singular points of $Σ$ with a unique tangent cone non-area minimizing on either side.
