Remarks on Semistable Points and Nonabelian Convexity of Gradient Maps

We study the action of a real reductive group $G$ on a Kahler manifold $Z$ which is the restriction of a holomorphic action of a complex reductive Lie group $U^{\mathbb{C}}\mathrm{.}$ We assume that the action of $U$, a maximal compact connected subgroup of $U^{\mathbb{C}}$ on $Z$ is Hamiltonian. If $G\subset U^{\mathbb{C}}$ is compatible, there is a corresponding gradient map ${\mu }_{\mathfrak{p}}:Z\to \mathfrak{p}$, where $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ is a Cartan decomposition of the Lie algebra of $G$. Our main results are the openness and connectedness of the set of semistable points associated with $G$-action on $Z$, a convexity theorem for the $G$-action on a $G$-invariant compact Lagrangian submanifold of $Z$, and a convexity result for two-orbit variety.