Minimal Lagrangian submanifolds in adjoint orbits and upper bounds on the first eigenvalue of the Laplacian

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概要

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• Let G be a compact semisimple Lie group, \mathfrak{g} its Lie algebra, (, ) an Ad<SUB>G</SUB>-invariant inner product on \mathfrak{g}, and M an adjoint orbit in \mathfrak{g}. In this article, if (M, (, )<SUB>|M</SUB>) is Kähler with respect to its canonical complex structure, then we give, for a closed minimal Lagrangian submanifold L⊂ M, upper bounds on the first positive eigenvalue λ<SUB>1</SUB>(L) of the Laplacian Δ<SUB>L</SUB>, which acts on C<SUP>∞</SUP>(L), and lower bounds on the volume of L. In particular, when (M, (, )<SUB>|M</SUB>) is Kähler-Einstein, ( ρ=comega, where ρ and ω are Ricci form and Kähler form of (M, (, )<SUB>|M</SUB>) with respect to the canonical complex structure respectively, and c is a positive constant, ) we prove λ<SUB>1</SUB>(L)≤ c. Combining with a result of Oh [{5}], we can see that L is Hamiltonian stable if and only if λ<SUB>1</SUB>(L)=c.

• 社団法人 日本数学会の論文
• 2003-01-01

著者

• Ono Hajime

  Department Of Mathematics Tokyo Institute Of Technology

• Ono Hajime

  Department Of Mathematics Faculty Of Science And Technology Tokyo University Of Science

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