Stability of minimal orbits
Author:
John E. Brothers
Journal:
Trans. Amer. Math. Soc. 294 (1986), 537552
MSC:
Primary 53C42; Secondary 49F22
MathSciNet review:
825720
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Abstract: Let be a compact Lie group of isometries of a riemannian manifold . It is well known that the minimal principal orbits are those on which the volume function , which assigns to the volume of the orbit of , is critical. It is shown that stability of a minimal orbit on which the hessian of is nonnegative is determined by the degree of involutivity of the distribution of normal planes to the orbits. Specifically, if the lengths of the tangential components of Lie brackets of pairs of orthonormal normal vector fields are sufficiently small relative to the hessian of , then the minimal orbit is stable, and conversely. Computable lower bounds are obtained for the values of these parameters at which stability turns to instability. These lower bounds are positive even in the case where is constant, and are finite unless the normal distribution is involutive. Several examples in which is a compact classical Lie group and is a subgroup of are discussed, showing in particular that the above estimates are sharp.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198608257203
PII:
S 00029947(1986)08257203
Keywords:
Minimal submanifold,
stable minimal submanifold,
stability,
orbit,
second variation of area
Article copyright:
© Copyright 1986
American Mathematical Society
