STABLE MINIMAL SUBMANIFOLDS IN COMPACT RANK ONE SYMMETRIC SPACES
STABLE MINIMAL SUBMANIFOLDS IN COMPACT RANK ONE SYMMETRIC SPACES
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DOI：
10.2748/tmj/1178228488
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发表时间：
1986
影响因子：
0.5
通讯作者：
Y. Ohnita
中科院分区：
文献类型：
--
作者：
Y. Ohnita
导论.浸入黎曼流形M中的紧致子流形M（无边界）称为极小子流形，如果对于M在M中的每一变形，其体积的第一变分为零。显然，如果M的体积是所有浸入中的局部极小，则M是M的极小子流形。但极小子流形的体积并不总是局部极小。现在我们知道了大量极小子流形的例子（例如全测地子流形、Kaehler流形的复子流形和紧变换群的极轨道等）。给定的极小子流形是否存在局部极小体积是一个重要的问题。我们说M中的紧致极小子流形M是稳定的，如果它的体积的第二变分对于M中的每一个变形都是非负的。显然，如果M有一个局部最小体积，那么它是稳定的。稳定极小子流形的类比一般极小子流形的类小得多。稳定极小子流形的存在性与周围流形的拓扑结构和黎曼结构密切相关。事实上，Simons [13]和Lawson-Simons [9]证明了以下显著定理。
Introduction. A compact submanifold M (without boundary) immersed in a Riemannian manifold M is called minimal if the first variation of its volume vanishes for every deformation of M in M. Clearly, if the volume of M is a local minimum among all immersions, M is a minimal submanifold of M. But the volume of a minimal submanifold is not always a local minimum. Nowadays we know a large number of examples of minimal submanifolds (e.g. totally geodesic submanifolds, complex submanifolds of Kaehler manifolds and extremal orbits of compact transformation groups, etc.). It is an important problem to know whether a given minimal submanifold has a local minimum volume or not. We say that a compact minimal submanifold M in M is stable if the second variation of its volume is nonnegative for every deformation of M in M. Clearly, if M has a local minimum volume, then it is stable. The class of stable minimal submanifolds is much smaller than the class of general minimal submanifolds. The existence of a stable minimal submanifold is closely related to the topological and Riemannian structures of the ambient manifold. In fact, Simons [13] and Lawson-Simons [9] proved the following remarkable theorems.