The volume minimizing problem of Lagrangian submanifolds
in K\"ahler manifolds under Hamiltonian deformations
was investigated first by Y. G. Oh about the beginning of 1990's.
It is fundamental and interesting as a geometric variational problem
related to Lagrangian submanifolds in
In this talk we shall discuss a nice classe of
Lagrangian submanifolds in complex hyperquadrics
and their Hamiltonian stability problems.
The complex hyperquadric is an Hermitian symmetric space of compact type and
rank $2$ :
$\widetilde{\mathrm Gr}_{2}({\bold R}^{n+2})
\cong Q_{n}({\bold C})
\cong SO(n+2)/SO(2)\times SO(n)$.
The relationship of minimal Lagrangian submanifold in complex hyperquadrics
with isoparametric hypersurfaces in spheres will be emphasized.
We shall mention our recent results on the Hamilitonian stability/instability
of compact minimal Lagrangian submanifold embedded in complex hyperquadrics
obtained as the images of the Gauss maps of
homogeneous isoparametric hypersurfaces in spheres.
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