A Crystalline Approximation Theorem for Hypersurfaces
A Crystalline Approximation Theorem for Hypersurfaces
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超曲面的晶体近似定理
DOI：
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发表时间：
1990
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通讯作者：
H. Huppertz
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作者：
H. Huppertz
积分流的紧性保证了在给定边界上存在面积最小化超曲面。然而，这样的方法无法明确地找到曲面。在这里，我们接近的问题，通过最大流和最小割之间的对偶找到一个绝对最小值。我们证明了一个新的“晶体近似”定理，表明任何超曲面可以近似的多边形与限制的方向，即从一定的细胞复杂的小平面。这使我们能够近似连续的最小曲面问题的一个有限网络上的流问题。这种线性规划问题的特殊算法使其具有实用性。准备证明我们的近似定理，我们给出了一个改进的证明贝西科维奇覆盖定理，有关的球包装。我们还开始了研究的多重性领域覆盖，往往是有用的扩展本地分析结果。我们给出了一个新的分析算法的最小成本循环，在极小曲面。我们证明了几个改进版本的三次变形定理，一个标准的结果在几何测度理论。然后，我们回顾了低维区域最小化曲面的曲率边界的一些结果，并为计算显式常数提供了足够的细节。我们的主要定理是使用一种新的技术，以取代任意电流设置边界（在一个适当的覆盖空间）。然后，我们采取的水平集的特征函数的光滑，以获得近似的表面具有一定的曲率界限。反过来，这可以通过来自多重网格（我们使用的单元复合体）的小平面来近似，仍然没有增加太多的质量。因此，特别是，任何边界上的面积最小化表面在我们的细胞复合体中具有良好的近似，并且我们可以通过我们的流算法快速找到。一些早期的技术已经能够计算面积最小化的表面上的边界是极端的，但我们是第一个处理任意边界循环，如打结曲线。许多极小曲面的结果对于面积以外的表面能（如椭圆和结晶积体）更难证明，但我们的构造对所有这些能量都同样有效。
The existence of an area-minimizing oriented hypersurface spanning a given boundary is guaranteed by compactness properties of the integral currents. Such methods, however, give no way to explicitly find the surface. Here we approach the problem of finding an absolute minimum through the duality between maximum flows and minimum cuts. We prove a new “crystalline approximation” theorem showing any hypersurface can be approximated by polygons with restricted orientations, namely facets from a certain cell complex. This allows us to approximate the continuous minimum surface problem by a flow problem on a finite network. Special algorithms for this linear programming problem make it practical to solve. Preparing to prove our approximation theorem, we give an improved proof of the Besicovitch covering theorem, related to sphere packings. We also initiate a study of the multiplicity of sphere coverings, often useful for extending local analytic results. We give a new analysis of algorithms for minimum-cost circulations, in terms of minimal surfaces. We prove several improved versions of the cubic deformation theorem, a standard result in geometric measure theory. Then we review some results bounding the curvature of areaminimizing surfaces in low dimensions, and we supply enough details to compute explicit constants. Our main theorem is obtained using a new technique to replace arbitrary currents by set boundaries (in an appropriate covering space). We then take a level set of a smoothing of the characteristic function, to get an approximation to the surface with certain curvature bounds. This can, in turn, be approximated by facets from a multigrid (the cell complex we use), still without much increase in mass. Thus, in particular, the area-minimizing surface on any boundary has a good approximation in our cell complex, and we can find this quickly by our flow algorithm. Some earlier techniques have been able to compute area-minimizing surfaces on boundaries which are extreme, but ours is the first to handle arbitrary boundary cycles, like knotted curves. Many minimal surface results are much harder to prove for surface energies other than area (like elliptic and crystalline intergrands), but our construction works equally well for all such energies.