标题： Schoen--Simon--Yau 和 Schoen--Simon 定理的迭代方法扩展，类似于 De Giorgi 的方法 显示英文标题

标题： Extensions of Schoen--Simon--Yau and Schoen--Simon theorems via iteration à la De Giorgi

摘要： 我们给出Schoen--Simon--Yau曲率估计及其相关的Bernstein型定理（1975）的另一种证明，并通过包括$6$维（稳定极小）浸入的情况来扩展原始结果。 关键步骤是一个$\epsilon$正则性定理，该定理假设第二基本形式的尺度不变$L^2$范数足够小。 此外，我们得到一个图描述，以Lipschitz多值的方式，对于任何维度为$n\geq 2$的稳定极小浸入，其可能具有局部有限$\mathcal{H}^{n-2}$测度的奇异集$\Sigma$，并且弱接近于一个超平面。 （事实上，如果$\mathcal{H}^{n-2}(\Sigma)=0$，结论可加强为光滑图的并集。） 这直接来自于一个$\epsilon$正则性定理，该定理假设尺度不变的$L^2$倾斜过度具有小性（当超曲面弱接近于一个超平面时得到验证）。 将多值分解专门应用于嵌入的情况，我们恢复了 Schoen--Simon 定理（1981）。 在两种$\epsilon$正则性定理中，相关量（分别为第二基本形式的长度和倾斜函数）在浸入的极小超曲面上解一个非线性偏微分方程。 证明是通过实施类似于 De Giorgi 的迭代方法（来自线性 De Giorgi--Nash--Moser 理论）来内在地进行的（不线性化偏微分方程）。 稳定性给出估计（内在的弱 Caccioppoli 不等式），这些估计尽管在非线性框架下也使迭代有效。 （在两种$\epsilon$正则性定理中，该方法给出了量化所需小性的显式常数。） 显示英文摘要

摘要： We give an alternative proof of the Schoen--Simon--Yau curvature estimates and associated Bernstein-type theorems (1975), and extend the original result by including the case of $6$-dimensional (stable minimal) immersions. The key step is an $\epsilon$-regularity theorem, that assumes smallness of the scale-invariant $L^2$ norm of the second fundamental form. Further, we obtain a graph description, in the Lipschitz multi-valued sense, for any stable minimal immersion of dimension $n\geq 2$, that may have a singular set $\Sigma$ of locally finite $\mathcal{H}^{n-2}$-measure, and that is weakly close to a hyperplane. (In fact, if $\mathcal{H}^{n-2}(\Sigma)=0$, the conclusion is strengthened to a union of smooth graphs.) This follows directly from an $\epsilon$-regularity theorem, that assumes smallness of the scale-invariant $L^2$ tilt-excess (verified when the hypersurface is weakly close to a hyperplane). Specialising the multi-valued decomposition to the case of embeddings, we recover the Schoen--Simon theorem (1981). In both $\epsilon$-regularity theorems the relevant quantity (respectively, length of the second fundamental form and tilt function) solves a non-linear PDE on the immersed minimal hypersurface. The proof is carried out intrinsically (without linearising the PDE) by implementing an iteration method \`{a} la De Giorgi (from the linear De Giorgi--Nash--Moser theory). Stability implies estimates (intrinsic weak Caccioppoli inequalities) that make the iteration effective despite the non-linear framework. (In both $\epsilon$-regularity theorems the method gives explicit constants that quantify the required smallness.)