  Let $(Z,\omega)$ be a \Keler manifold and let $U$ be a compact connected Lie
group with Lie algebra $\mathfrak{u}$ acting on $Z$ and preserving $\omega$. We
assume that the $U$-action extends holomorphically to an action of the
complexified group $U^{\mathbb C}$ and the $U$-action on $Z$ is Hamiltonian.
Then there exists a $U$-equivariant momentum map $\mu : Z\to \mathfrak{u}$. If
$G\subset U^{\mathbb C}$ is a closed subgroup such that the Cartan
decomposition $U^{\mathbb C} = U\text{exp}(i\mathfrak{u})$ induces a Cartan
decomposition $G = K\text{exp}(\mathfrak{p}),$ where $K = U\cap G$,
$\mathfrak{p} = \mathfrak{g}\cap i\mathfrak{u}$ and $\mathfrak{g}=\mathfrak k
\oplus \mathfrak p$ is the Lie algebra of $G$, there is a corresponding
gradient map $\mu_\mathfrak{p} : Z\to \mathfrak{p}$. If $X$ is a $G$-invariant
compact and connected real submanifold of $Z,$ we may consider $\mu_{\mathfrak
p}$ as a mapping $\mu_\mathfrak{p} : X\to \mathfrak{p}.$ Given an
$\mathrm{Ad}(K)$-invariant scalar product on $\mathfrak p$, we obtain a Morse
like function $f=\frac{1}{2}\parallel \mu_{\mathfrak p} \parallel^2$ on $X$. We
point out that, without the assumption that $X$ is real analytic manifold, the
Lojasiewicz gradient inequality holds for $f$. Therefore the limit of the
negative gradient flow of $f$ exists and it is unique. Moreover, we prove that
any $G$-orbit collapses to a single $K$-orbit and two critical points of $f$
which are in the same $G$-orbit belong to the same $K$-orbit. We also
investigate convexity properties of the gradient map $\mu_\mathfrak{p}$ in the
Abelian cases. In particular, we study two orbits variety $X$ and we
investigate topological and cohomological properties of $X$.
