发表机构
University College London; The Chinese University of Hong Kong; University of Warwick; University of Toronto（伦敦大学学院; 香港中文大学; 华威大学; 多伦多大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立了稳定自由边界极小超曲面的Schoen-Simon正则性与紧性理论，通过倾斜函数的ε-正则性和De Giorgi迭代证明层片定理，并推广了自由边界Almgren-Pitts存在性理论到任意维数。
AI 中文摘要
我们建立了Schoen-Simon关于稳定极小超曲面正则性与紧性理论的自由边界版本，该理论适用于完备的$(n+1)$维带边黎曼流形，遵循第一作者引入的内蕴PDE方法。关键步骤是在局部弱平坦性可能出现在$\partial M$上某点附近的两种可能情形下证明层片定理：相对于$\partial M$的平坦性，以及与$\partial M$正交相交的带边参考超曲面的平坦性。我们通过几何适应的倾斜函数的$\varepsilon$-正则性结果获得这些定理，这些倾斜函数量化平坦性同时跟踪环境几何。这些倾斜函数满足超曲面上的非线性椭圆PDE，可通过De Giorgi迭代进行内蕴分析，这得益于由稳定性得到的弱Caccioppoli型不等式。两个结果均适用于具有局部有限$(n-2)$维Hausdorff测度奇异集的浸入超曲面。当特化为嵌入时，它们产生紧性与正则性：具有一致有界面积和局部有限$(n-2)$维测度奇异集的稳定自由边界嵌入极小超曲面序列，存在一个子序列收敛到一个自由边界极小超曲面，该超曲面在Hausdorff维数至多$n-7$的奇异集之外光滑嵌入。收敛在该奇异集之外是光滑且图形的，可能具有重数。作为应用，我们将第二作者与X. Zhou发展的自由边界Almgren-Pitts存在性理论推广到任意维数，并证明每个$n+1$维紧致带边黎曼流形包含一个自由边界极小超曲面，其在Hausdorff维数至多$n-7$的奇异集之外光滑嵌入。
英文摘要
We establish the free-boundary version of Schoen-Simon's regularity and compactness theory for stable minimal hypersurfaces in a complete $(n+1)$-dimensional Riemannian manifold-with-boundary, following the intrinsic PDE scheme introduced by the first-named author. The key step is to prove sheeting theorems in the two possible scenarios in which local weak flatness can occur near a point on $\partial M$: flatness with respect to $\partial M$, and flatness with respect to a reference hypersurface-with-boundary meeting $\partial M$ orthogonally. We obtain these theorems via $\varepsilon$-regularity results for geometrically adapted tilt functions that quantify flatness while keeping track of the ambient geometry. These tilt functions satisfy nonlinear elliptic PDEs on the hypersurface which are amenable to an intrinsic analysis via De Giorgi iteration, made possible by the weak Caccioppoli-type inequalities obtained from stability. Both results apply to immersed hypersurfaces with a singular set of locally finite $(n-2)$-dimensional Hausdorff measure. When specialised to embeddings, they yield compactness and regularity: a sequence of stable free-boundary embedded minimal hypersurfaces with uniformly bounded area and singular sets of locally finite $(n-2)$-measure admits a subsequence converging to a free-boundary minimal hypersurface that is smoothly embedded away from a singular set of Hausdorff dimension at most $n-7$. Convergence is smooth and graphical, possibly with multiplicity, away from this singular set. As an application, we extend to arbitrary dimension the free-boundary Almgren-Pitts existence theory developed by the second-named author with X. Zhou and prove that every compact Riemannian manifold-with-boundary of dimension $n+1$ contains a free-boundary minimal hypersurface smoothly embedded outside a singular set of Hausdorff dimension at most $n-7$.
