New applications of mapping degrees to minimal surface theory
New applications of mapping degrees to minimal surface theory
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映射度在最小曲面理论中的新应用
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发表时间：
1989
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通讯作者：
B. White
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作者：
B. White
在[21]中，Tomi和Tromba展示了如何使用Smale [19]的度理论来解决长期开放的问题，即证明R 3的凸子集的边界中的每一条光滑嵌入曲线必须绑定嵌入极小圆盘。后来Almgren和Simon [4]以及Meeks和Yau [15]给出了不同的证明。本文给出了度理论在极小曲面上的其他应用。特别是，我们显示：（1）若Φ是R3中的偶常系数参数椭圆泛函，η是R的严格凸子集边界上的光滑嵌入曲线，则η有界于嵌入的Φ-平稳和Φ-稳定圆盘.此外，一般的这样的曲线界定了奇数个嵌入的Φ静止圆盘和偶数个嵌入的Φ-静止表面的每个其他拓扑类型。(2)设TV是一个光滑的黎曼3-流形，其严格平均凸边界与2-球面同构。设N不与3-球同胚，或N包含一个无边界的紧致极小曲面。则N中存在一个嵌入极小圆盘序列Di，使得dDi C9 JV，dDi收敛于光滑的嵌入曲线7，且Di的面积趋于无穷大. (3)在Rn中存在一个完备极小超曲面M，使得M是奇异的，M不是锥，并且M在oo渐近于一个除原点外正则的面积极小锥C. (4)在Rn中存在一个完备的面积极小化超曲面M，使得M渐近于一个除原点外正则的面积极小化锥C，但M不全等于与C相关联的极小超曲面的叶状的任何叶.这些结果分别在§§2，3，4和δ中得到证明。所有这些都依赖于§1中的附录，§5是§4的延续，但除此之外，各节都是相互独立的。第6章举例说明
In [21], Tomi and Tromba showed how it was possible to use the degree theory of Smale [19] to solve the long open problem of proving that every smooth embedded curve in the boundary of a convex subset of R 3 must bound an embedded minimal disk. Later Almgren and Simon [4] and Meeks and Yau [15] gave different proofs. In this paper we give other applications of degree theory to minimal surfaces. In particular, we show: (1) If Φ is an even constant coefficient parametric elliptic functional in R 3 and η is a smooth embedded curve on the boundary of a strictly convex subset of R , then η bounds an embedded Φ-stationary and Φ-stable disk. Furthermore, a generic such curve bounds an odd number of embedded Φstationary disks and an even number of embedded Φ-stationary surfaces of each other topological type. (2) Let TV be a smooth Riemannian 3-manifold with strictly mean convex boundary diffeomorphic to the 2-sphere. Suppose either that N is not diffeomorphic to the 3-ball, or else that N contains a compact minimal surface without boundary. Then there exists a sequence Di of embedded minimal disks in N such that dDi C 9JV, dDi converges to a smooth embedded curve 7, and the area of Di tends to infinity. (3) There exists a complete minimal hypersurface M in R n such that M is singular, M is not a cone, and M is asymptotic at oo to an area minimizing cone C that is regular except at the origin. (4) There exists a complete area minimizing hypersurface M in R n such that M is asymptotic to an area minimizing cone C that is regular except at the origin, but M is not congruent to any leaf of the foliation of minimal hypersurfaces associated with C. These results are proved in §§2, 3, 4, and δ, respectively. All depend on the preliminaries in §1, and §5 Is a continuation of §4, but otherwise the sections are independent of each other. §6 discusses examples.