The equivariant Dehn's lemma and loop theorem
The equivariant Dehn's lemma and loop theorem
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等变 Dehn 引理和循环定理
DOI：
10.1007/bf02566211
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发表时间：
1981
影响因子：
0.9
通讯作者：
S. Yau
中科院分区：
文献类型：
--
作者：
W. H. Meeks;S. Yau
在[4]中，作者观察到三维流形理论中的拓扑方法可以修改，以解决经典欧几里得空间极小曲面理论中的一些老问题（参见[1]，[12]）。在b[4]和[5]中，我们发现我们可以用最小曲面理论来推广Papakriakopoulous， Whitehead和Shapiro， Stalling和Epstein关于Dehn引理，循环定理和球体定理的定理。我们研究这些拓扑定理的关键是：给定一个映射到三维流形M中的盘或球的映射族，我们最小化这个族中的映射的面积（相对于拉回的度量），并证明最小映射的存在性。然后利用图的面积最小化性质和拓扑中的塔结构，证明了族中任何面积最小化的图都是一个嵌入。这样，我们就用极小曲面实现了上述拓扑定理的解。在[4]和[5]中，我们利用上述面积最小化解证明了环和球定理的等变版本，并将这些新定理应用于[11]中r3上紧群作用的分类。本文通过证明给定属的最小面积嵌入平面域的存在性和最小面积平面域的不相交性，将[4]和[5]中的一些定理推广到紧平面域。然后利用这个不相交性质证明了平面域的等变Dehn引理。另一方面，我们使用不同的变分方法来得到环定理的测地线版本。更确切地说，我们证明了：假设包含边界的诱导映射i.:~rl（OM）—*”rr~(M)具有非平凡核K，那么对于OM上的任何度量，K中任何长度最小的非平凡测地线都嵌入，并且任意两个测地线相等或不相交。这个测地线环定理与上述等变Dehn引理结合，在[5]中得到了等变环定理的一个新版本。由于曲线在曲面上的放置更容易理解，这个新的等变环定理更容易理解
In [4] the authors observed that the topological methods in the theory of three-dimensional manifolds can be modified to settle some old problems in the classical theory of minimal surfaces in euclidean space (see also [1], [12]). In [4] and [5] we found that we could use the theory of minimal surfaces to extend the theorems of Papakriakopoulous, Whitehead and Shapiro, Stalling and Epstein on the Dehn's lemma, loop theorem and sphere theorem. The key point to our approach to these topological theorems is the following: Given a certain family of maps of the disk or sphere into our three-dimensional manifold M, we minimize the area of the maps (with respect to the pulled back metric) in this family and prove the existence of the minimal map. Then by using the area minimizing property of the map and the tower construction in topology, we prove that any area minimizing map in the family is an embedding. In this way, we realize the solutions to the above topological theorems by minimal surfaces. In [4] and [5] we used the above area minimizing solutions to prove equivariant versions of the loop and the sphere theorem, and we applied these new theorems to the classification of compact group actions on R 3 in [11]. In this paper we generalize some of the theorems in [4] and [5] to compact planar domains by proving the existence of embedded planar domains of least area of a given genus and by proving a certain disjointness property for planar domains of least area. We then use this disjointness property to prove the equivariant Dehn's lemma for planar domains. On the other hand, we use a different variation approach to get a geodesic version of the loop theorem. More precisely, we prove the following: suppose that the induced map i.:~rl(OM)--* "rr~(M) of the inclusion of the boundary has nontrivial kernel K. Then for any metric on OM, any nontrivial geodesic of least length in K is embedded and any two such geodesics are equal or disjoint. This geodesic loop theorem coupled with the above equivariant Dehn's lemma yields a new version of the equivariant loop theorem in [5]. As the placement of curves on a surface is easier to understand this new equivariant loop theorem is easier to