I am studying the paper by E. Lerman: https://arxiv.org/pdf/math/0410568.pdf

Let $ (M,\sigma)$ be a connected symplectic manifold with an hamiltonien action of a compact Lie group $ G$ , so that there exist a moemnt map $ $ \mu : M\to\mathcal{G}^\ast$ $ $ \mathcal{G}^\ast$ being the dual of the Lie algebra of $ G$ . We assume that $ \mu$ is $ G$ -equivariant: $ $ \mu(g\cdot x)=\mathrm{Ad}_g^\ast\circ\mu(x)$ $ and that $ \mu$ is proper (the preimage of any compact is compact). Let $ f=\|\mu\|^2$ (for an Ad-invariant norm on $ \mathcal{G}^\ast$ ).

I know that the moment map is important by:

1- a convexity theorem of Atiyah and Guillemin-Sternberg.

2- symplectic reduction, where the quotient of the zero level of the moment map by the group makes it possible to construct new symplectic manifolds.

Hence my question:

What is the motivation to study the norm squared of a moment map? in particular, why is it important to know that the zero level set of the moment map is a retract by deformation of a piece of the manifold?

As I understand it, $ f$ behaves like a Morse-Bott function (Kirwan works) and that the stable manifold of a critical component of $ f$ is a submanifold. That the gradient flow of $ f$ is defined for all $ t\geq0$ . Here Lerman asserts that this is true because $ f$ is proper, but $ x^3$ is proper but its gradient $ -3x^2\partial_x$ is not defined for all $ t\geq0$ .

I think we have to show that $ \nabla_f$ is $ G$ -invariant and therefore complete.

That $ f$ is real analytic to show that the limit of a trajectory of any point $ \phi_t(x)$ is reduced to a point $ \phi_\infty(x)$ . That the applications $ t\to\phi_t(x)$ and $ x\to\phi_\infty(x)$ are continuous.