Two-dimensional area minimizing integral currents are classical minimal surfaces
Two-dimensional area minimizing integral currents are classical minimal surfaces
复制标题
二维面积最小化积分电流是经典的最小表面
DOI：
10.1090/s0894-0347-1988-0946554-0
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发表时间：
1988
影响因子：
3.9
通讯作者：
Sheldon Chang
中科院分区：
文献类型：
--
作者：
Sheldon Chang
几何测度论保证了紧致黎曼流形上存在面积最小的跨越给定边界的积分流或表示给定的积分同调类。我们研究了这种广义曲面的正则性。证明了当积分流极小化区域的维度为2时，它们是经典的极小曲面。在这个正则性结果的结果中，我们现在知道，紧致黎曼流形上的任何二维积分同调类都可以由只有有限多个交点的经典闭极小曲面的有限积分线性组合来表示。利用F.Almgren在[A]中发展的多值函数理论证明了这一结果。我们推广了他的文章中的许多重要估计，并推广了他的中心流形的构造。我们利用多值函数中的分支中心流形和最小电流面积的最低阶项来构造内部奇点附近的两个分支曲面序列，以逐步分离附近的奇点。通过本文的分析，我们可以得出结论：广义曲面一定与其中一个分支曲面重合。马萨诸塞州剑桥市哈佛大学数学系02138许可或版权限制可能适用于再分发；请参阅https://www.ams.org/journal-terms-of-use
Geometric measure theory guarantees the existence of area minimizing integral currents spanning a given boundary or representing a given integral homology class on a compact Riemannian manifold. We study the regularity of such generalized surfaces. We prove that in case the dimension of the area minimizing integral currents is two, then they are classical minimal surfaces. Among the consequences of this regularity result, we know now that any two dimensional integral homology class on a compact Riemannian manifold can be represented by a finite integral linear combination of classical closed minimal surfaces that have only finitely many intersection points. The result is proved by using the theory of multiple-valued functions developed by F. Almgren in [A]. We extend many important estimates in his paper and extend his construction of center manifolds. We use the branched center manifolds and lowest order term in the multiple-valued functions approximating the area minimizing currents to construct two sequences of branched surfaces near an interior singular point to separate the nearby singularity gradually. The analysis developed in this paper enables us to conclude the generalized surface must coincide with one of the branched surfaces. DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY, CAMBRIDGE, MASSACHUSETTS 02138 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use