论文标题
由环境函数规定的平均曲率的高空曲面：紧凑性结果
Hypersurfaces with mean curvature prescribed by an ambient function: compactness results
论文作者
论文摘要
首先，我们考虑具有本地有限周围的集合的界限类别（弱定义的）平均曲率为$gν$，对于给定的连续积极环境函数$ g $，其中$ν$表示内部正常。众所周知，由于“隐藏边界”的出现，即在该类别内的varifolds意义上采取限制是不可能的，也就是说，（较弱定义的）平均曲率消失的部分（正值均匀度量），因此$ g $不会在极限下的平均曲率处方。作为更普遍结果的一个特殊情况，我们证明（（弱定义）第二个基本形式的本地均匀$ l^q $ bounds（以$ q>> 1 $）为$>> 1 $，除了周围的习惯本地统一界，还导致了$ g $均值的弯曲率的紧凑型边界类别。 证明依赖于将边界视为“方向的积分varifolds”，以利用其优势性特征（当将其视为（无定向）varifolds时丢失）。具体而言，它依赖于定向积分varifolds的曲率系数弱概念的制定和分析（灵感来自Hutchinson的工作）。 This framework gives (with no additional effort) a compactness result for oriented integral varifolds with curvature locally bounded in $L^q$ with $q>1$ and with mean curvature prescribed by any $g\in C^0$ (in fact, the function can vary with the varifold for which it prescribes the mean curvature, as long as there is locally uniform convergence of the prescribing functions).我们的概念和陈述是在Riemannian的歧管中给出的，而无需作为边界出现的定向的Varifold（例如，它们可能来自双面沉浸式）。
We consider, in a first instance, the class of boundaries of sets with locally finite perimeter whose (weakly defined) mean curvature is $g ν$, for a given continuous positive ambient function $g$, and where $ν$ denotes the inner normal. It is well-known that taking limits in the sense of varifolds within this class is not possible in general, due to the appearence of "hidden boundaries", that is, portions (of positive measure with even multiplicity) on which the (weakly defined) mean curvature vanishes, so that $g$ does not prescribe the mean curvature in the limit. As a special instance of a more general result, we prove that locally uniform $L^q$-bounds on the (weakly defined) second fundamental form, for $q>1$, in addition to the customary locally uniform bounds on the perimeters, lead to a compact class of boundaries with mean curvature prescribed by $g$. The proof relies on treating the boundaries as 'oriented integral varifolds', in order to exploit their orientability feature (that is lost when treating them as (unoriented) varifolds). Specifically, it relies on the formulation and analysis of a weak notion of curvature coefficients for oriented integral varifolds (inspired by Hutchinson's work). This framework gives (with no additional effort) a compactness result for oriented integral varifolds with curvature locally bounded in $L^q$ with $q>1$ and with mean curvature prescribed by any $g\in C^0$ (in fact, the function can vary with the varifold for which it prescribes the mean curvature, as long as there is locally uniform convergence of the prescribing functions). Our notions and statements are given in a Riemannian manifold, with the oriented varifolds that need not arise as boundaries (for instance, they could come from two-sided immersions).
