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, $g$ its Lie algebra, $(,)$ an $A{d}_G$-invariant inner product on $g$, and $M$ an adjoint orbit in 9. In this article, if $(M,(,{)}_{|M})$ is Kähler with respect to its canonical complex structure, then we give, for a closed minimal Lagrangian submanifold $L\subset M$, upper bounds on the first positive eigenvalue ${\lambda }_1\left(L\right)$ of the Laplacian ${\Delta }_L$, which acts on $C^{\infty }\left(L\right)$, and lower bounds on the volume of $L$. In particular, when $(M,(,{)}_{|M})$ is Kähler-Einstein, ($p=c\omega$, where $p$ and $\omega$ 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 ${\lambda }_1\left(L\right)\le c$. Combining with a result of Oh [5], we can see that $L$ is Hamiltonian stable if and only if ${\lambda }_1\left(L\right)=c$.