Stability of minimal orbits

Author:
John E. Brothers

Journal:
Trans. Amer. Math. Soc. **294** (1986), 537-552

MSC:
Primary 53C42; Secondary 49F22

DOI:
https://doi.org/10.1090/S0002-9947-1986-0825720-3

MathSciNet review:
825720

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[**David Bindschadler,**BD**]*Invariant solutions to the oriented Plateau problem of maximal codimension*, Trans. Amer. Math. Soc.**261**(1980), no. 2, 439–462. MR**580897**, https://doi.org/10.1090/S0002-9947-1980-0580897-7**[**John E. Brothers,**BJ1**]*Invariance of solutions to invariant parametric variational problems*, Trans. Amer. Math. Soc.**262**(1980), no. 1, 159–179. MR**583850**, https://doi.org/10.1090/S0002-9947-1980-0583850-2**[**-,**BJ2**]*Second variation estimates for minimal orbits*, Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, R.I., 1985.**[**Herbert Federer,**F**]*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****[**Wu-yi Hsiang,**H**]*On the compact homogeneous minimal submanifolds*, Proc. Nat. Acad. Sci. U.S.A.**56**(1966), 5–6. MR**0205203****[**Wu-yi Hsiang and H. Blaine Lawson Jr.,**HL**]*Minimal submanifolds of low cohomogeneity*, J. Differential Geometry**5**(1971), 1–38. MR**0298593****[**Shoshichi Kobayashi and Katsumi Nomizu,**KN**]*Foundations of differential geometry. Vol I*, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR**0152974****[**H. Blaine Lawson Jr.,**L**]*Lectures on minimal submanifolds. Vol. I*, 2nd ed., Mathematics Lecture Series, vol. 9, Publish or Perish, Inc., Wilmington, Del., 1980. MR**576752****[**H. Blaine Lawson Jr. and James Simons,**LS**]*On stable currents and their application to global problems in real and complex geometry*, Ann. of Math. (2)**98**(1973), 427–450. MR**0324529**, https://doi.org/10.2307/1970913**[**Barrett O’Neill,**ON**]*The fundamental equations of a submersion*, Michigan Math. J.**13**(1966), 459–469. MR**0200865****[**James Simons,**S**]*Minimal varieties in riemannian manifolds*, Ann. of Math. (2)**88**(1968), 62–105. MR**0233295**, https://doi.org/10.2307/1970556**[**Dao Chong Thi,**T**]*Minimal real currents on compact Riemannian manifolds*, Math. USSR Izv.**11**(1977), 807-820.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
53C42,
49F22

Retrieve articles in all journals with MSC: 53C42, 49F22

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0825720-3

Keywords:
Minimal submanifold,
stable minimal submanifold,
stability,
orbit,
second variation of area

Article copyright:
© Copyright 1986
American Mathematical Society