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

Abstract

Let G be a compact semisimple Lie group, \mathfrak{g} its Lie algebra, (, ) an Ad_G-invariant inner product on \mathfrak{g}, and M an adjoint orbit in \mathfrak{g}. In this article, if (M, (, )_|M) 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 λ₁(L) of the Laplacian Δ_L, which acts on C^∞(L), and lower bounds on the volume of L. In particular, when (M, (, )_|M) is Kähler-Einstein, ( ρ=comega, where ρ and ω are Ricci form and Kähler form of (M, (, )_|M) with respect to the canonical complex structure respectively, and c is a positive constant, ) we prove λ₁(L)≤ c. Combining with a result of Oh [{5}], we can see that L is Hamiltonian stable if and only if λ₁(L)=c.