Regularity of minimal hypersurfaces with a common free boundary
Regularity of minimal hypersurfaces with a common free boundary
复制标题
具有共同自由边界的最小超曲面的正则性
DOI：
--
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
B. Krummel
中科院分区：
文献类型：
--
作者：
B. Krummel
设$$N$$ N是一个黎曼流形，并考虑三个或更多个的平稳联合$$C^{1,mu }$$ C1，μ超曲面-带边界$$M_k subset N$$ Mk∧N有一个公共边界$$Gamma $$ Γ。我们证明，如果$$N$$ N是光滑的，那么$$Gamma $$ Γ是光滑的，并且每个$$M_k$$ Mk平滑到$$Gamma $$ Γ（在$$N$$ N是实解析的情况下是实解析的）。因此，我们加强了Wickramasekera关于稳定协维1积分变形正则性的结论，得出在$$V$$ V是$$(n+1)$$ （n+1）维的平稳、稳定、积分$$n$$ n-变形的更强假设下，光滑的（实解析的）黎曼流形，使得$$Vert VVert $$‖V‖的支持在任何地方都不是三个或更多光滑的（实解析的）带边界的超曲面沿公共边界会合的并，如果$$n le 6$$ n≤6,$$V$$ V的奇异集是空的，如果$$n = 7$$ n=7，则是离散的，如果$$n ge 8$$ n≥8，则最多有$$n-7$$ n-7的豪斯多夫维数。
Let $$N$$N be a Riemannian manifold and consider a stationary union of three or more $$C^{1,mu }$$C1,μ hypersurfaces-with-boundary $$M_k subset N$$Mk⊂N with a common boundary $$Gamma $$Γ. We show that if $$N$$N is smooth, then $$Gamma $$Γ is smooth and each $$M_k$$Mk is smooth up to $$Gamma $$Γ (real analytic in the case $$N$$N is real analytic). Consequently we strengthen a result of Wickramasekera for stable codimension 1 integral varifolds regularity to conclude that under the stronger hypothesis that $$V$$V is a stationary, stable, integral $$n$$n-varifold in an $$(n+1)$$(n+1)-dimensional, smooth (real analytic) Riemannian manifold such that the support of $$Vert VVert $$‖V‖ is nowhere locally the union of three or more smooth (real analytic) hypersurfaces-with-boundary meeting along a common boundary, the singular set of $$V$$V is empty if $$n le 6$$n≤6, is discrete if $$n = 7$$n=7, and has Hausdorff dimension at most $$n-7$$n-7 if $$n ge 8$$n≥8.