Let be a compact semisimple Lie group, its Lie algebra, an -invariant inner product on , and an adjoint orbit in 9. In this article, if is Kähler with respect to its canonical complex structure, then we give, for a closed minimal Lagrangian submanifold , upper bounds on the first positive eigenvalue of the Laplacian , which acts on , and lower bounds on the volume of . In particular, when is Kähler-Einstein, (, where and are Ricci form and Kähler form of with respect to the canonical complex structure respectively, and is a positive constant,) we prove . Combining with a result of Oh , we can see that is Hamiltonian stable if and only if .
Hajime ONO. "Minimal Lagrangian submanifolds in adjoint orbits and upper bounds on the first eigenvalue of the Laplacian." J. Math. Soc. Japan 55 (1) 243 - 254, January, 2003. https://doi.org/10.2969/jmsj/1196890852