Hypersurfaces with mean curvature prescribed by an ambient function: compactness results
Hypersurfaces with mean curvature prescribed by an ambient function: compactness results
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DOI：
10.1016/j.jfa.2023.110125
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发表时间：
2022-12
影响因子：
1.7
通讯作者：
C. Bellettini
中科院分区：
文献类型：
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作者：
C. Bellettini
首先，我们考虑具有局部有限周长的集合的边界类，对于给定的连续正环境函数g，其（弱定义的）平均曲率为gν，其中ν表示内法线。这是众所周知的，采取限制在这个类的varifolds的意义上是不可能的，一般由于外观的“隐藏的边界”，即部分（正措施，甚至多重性）的（弱定义）平均曲率消失，使g不规定的平均曲率的限制。作为更一般结果的一个特殊例子，我们证明了（弱定义的）第二基本形式上的局部一致Lq-界，当q> 1时，除了周长上通常的局部一致界之外，还导致了一类具有由g规定的平均曲率的紧边界。证明依赖于将边界视为定向积分变分，以利用其可定向性特征（当将其视为（无定向）变分时会丢失）。具体来说，它依赖于定向积分变分的曲率系数的弱概念的公式化和分析（受哈钦森的工作[7]的启发）。这个框架给出了（没有额外的努力）一个紧性结果，其曲率在L q中局部有界，q> 1，且平均曲率由任意g∈ C 0规定（事实上，函数可以随它规定平均曲率的变倍而变化，只要规定函数局部一致收敛）。我们的概念和陈述是在黎曼流形中给出的，其中定向变分不需要作为边界出现（例如，它们可以来自双侧浸入）。
We consider, in a first instance, the class of boundaries of sets with locally finite perimeter whose (weakly defined) mean curvature is gν, 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 appearance 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 [7]). 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∈ 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).