  We presented a Hilbert-Mumford criterion for polystablility associated with
an action of a real reductive Lie group $G$ on a real submanifold $X$ of a
Kahler manifold $Z$. Suppose the action of a compact Lie group with Lie algebra
$\mathfrak{u}$ extends holomorphically to an action of the complexified group
$U^\mathbb{C}$ and that the $U$-action on $Z$ is Hamiltonian. If $G\subset
U^\mathbb{C}$ is compatible, there is a corresponding gradient map
$\mu_\mathfrak{p}: X\to \mathfrak{p}$, where $\mathfrak{g} = \mathfrak{k}
\oplus \mathfrak{p}$ is a Cartan decomposition of the Lie algebra of $G$. Under
some mild restrictions on the $G$-action on $X,$ we characterize which
$G$-orbits in $X$ intersect $\mu_\mathfrak{p}^{-1}(0)$ in terms of the maximal
weight function, which we viewed as a collection of maps defined on the
boundary at infinity ($\partial_\infty G/K$) of the symmetric space $G/K$.
