Properties of Gradient maps associated with Action of Real reductive Group

TL;DR

This paper studies the properties of gradient maps associated with actions of real reductive groups on Kähler manifolds, establishing gradient flow convergence, orbit structure, and convexity properties.

Contribution

It introduces a Morse-like function based on the gradient map, proves the convergence of its gradient flow, and analyzes orbit and convexity properties in the context of real reductive group actions.

Findings

Gradient flow of the Morse-like function converges uniquely.

Any G-orbit collapses to a single K-orbit.

Critical points in the same G-orbit are in the same K-orbit.

Abstract

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 $X$.