Properties of Gradient maps associated with Action of Real reductive Group

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 $\mathfrak{g}=\mathfrak k \oplus \mathfrak p$istheLiealgebraof$G$, 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 $\mu_{\mathfrak p}$asamapping$\mu_\mathfrak{p} : X\to \mathfrak{p}.$Givenan$$\mathrm{A}\mathrm{d}$(K)$-invariantscalarproducton$\mathfrak p$, 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 $X$.