Spherical gradient manifolds

We study the action of a real-reductive group $G=K\exp (\text{lie}p)$ on real-analytic submanifold $X$ of a K"ahler manifold $Z$. We suppose that the action of $G$ extends holomorphically to an action of the complexified group $G^{\text{mbb}}C$ such that the action of a maximal Hamiltonian subgroup is Hamiltonian. The moment map $\mu$ induces a gradient map $\mu_\lie{p}\colon X\to\lie{p}$\mathrm{.}Weshowthat$\mu_\lie{p}$almostseparatesthe$K$–orbits if and only if a minimal parabolic subgroup of $G$ has an open orbit. This generalizes Brion’s characterization of spherical K"ahler manifolds with moment maps.