Remarks on the stability of minimal submanifolds of IRn
Remarks on the stability of minimal submanifolds of IRn
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DOI：
10.1007/bf01190946
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发表时间：
1975-06
影响因子：
0.8
通讯作者：
J. Spruck
中科院分区：
文献类型：
--
作者：
J. Spruck
设Mm是具有边界~M的m维紧致可定向C流形，x：m-->；IR“是M在欧氏空间中的极小浸入。众所周知，M相对于m维体积是稳定的。也就是说，如果E是M上消失在OM上的正规向量场，如果~p~表示E的流，则设A(T)=Volume~pt(M)，--d~tt_o=0。最近Barbosa和Do Carmo[-1]证明了如果M是IR3(m=2，n=3)中的极小曲面，使得M的球面像面积小于2，则M是稳定的。常量2~是尖锐的。本文的目的是在一般情况下证明类似但较弱的稳定性结果。
Let M m be an m-dimensional compact orientable C manifold with boundary~ M and let x: M---> IR" be a minimal immersion of M into Euclidean n space. It is well known that M is stationary with respect to m-dimensional volume. That is, if E is a normal vector field on M vanishing on OM and if~ p~ denotes the flow by E, then setting A (t)= volume~ pt (M),--d~ t t_o= 0. _ We say generated that M is (infinitesimally) stable if dt 2 t= o> 0, ie A (M) is a strict minimum for all such variations.Recently Barbosa and do Carmo [-1] have shown that if M is a minimal surface in IR 3 (m= 2, n= 3) such that the area of the spherical image (without multiplicity) of M is less than 2~, then M is stable. The constant 2~ is sharp. It is the purpose of this note to prove similar but weaker stability results in the general case.