How to use that the Hessian is negative definite in this proof Let $X$ be a  Riemannian compact manifold on which acts a compact Lie group $H^+$. Let $f^+ : X \rightarrow \mathbb{R} $ be a smooth function on $X$. Consider a Lie subgroup $U$ of $H^+$ and suppose that $U$ acts on the set of critical points of $f^+$.  Denote by $\phi $  the flow of the gradient vector field of $f^+$. Suppose that $\phi$ is $U$-invariant, and that $\lim_{t \rightarrow +\infty} \phi_t(x)$ exists and it  belongs to the set of critical points of $f^+$ for all $x \in X$.$\DeclareMathOperator{\Hess}{\operatorname{Hess}}$
Let $\alpha:= U.x_0$ be a $U$-orbit which pass through a critical point $x_0$ and denote by $\alpha^+:= H^+.x_0$ the corresponding $H^+$-orbit on $X$.  Let $C$ be a connected component of $\alpha$ and let $M_C$ be the set $M_C:= \lbrace x \in X, \lim_{t \rightarrow+ \infty} \phi_t(x) \in C \rbrace $.
$\textbf{Question}$: Show that if  flow  $\phi$ preserves the orbit $\alpha ^+$ and if the Hessian $\Hess_x(f^+)$  is negative definite on $T_x( \alpha^+)$ for all $x \in \alpha $,  then  $\alpha^+ \subset M_C $  in a neighborhood $V$ of $C$.
My motivaion to ask this question is to understand the proof of proposition 3.9 in the paper Matsuki correspondence for sheaves.

 A: The facts that the Hessian is negative definite on the normal bundle $T_\alpha (\alpha^+)$ to $\alpha$ in $\alpha^+$ and that $\alpha$ are all critical points tells you that there is a neighborhood $V$ of $C$ in $\alpha^+$ such that $V \subseteq M_C$. (In other words, near enough to $C$, the gradient flow will return you to $C$.) Of course, it follows that $\alpha^+ \subseteq M_C$ in that neighborhood.
(Note that $T_\alpha (\alpha^+)$ in (3.8.2.b) apparently refers to the normal bundle to $\alpha$ in $\alpha^+$. The Hessian is not negative definite across any tangent space $T_x (\alpha^+)$ at a point $x \in \alpha$, since it is zero over the directions tangent to $\alpha$, (3.8.2.a).)
A: The authors use the Morse-Bott theory for the equivariant moment map $f^+$. Any connected component $C$ of the critical set of $f^+$ is a submanifold and for any critical point $x\in C$, $T_xC=\ker\mathrm{Hess}_x(f^+)$. Moreover
$$T_xX=T_xC\oplus E_x^-\oplus E_x^+$$
where $E_x^-$ is spanned by the negative eigenspaces and $E_x^+$ is spanned by the positive eigenspaces  of $\mathrm{Hess}_x(f^+)$ (the hessian is symmetric, thus all its eigenvalues are real).
If $U$ is connected, then $U\cdot x\subset C$ and since $X$ is compact, it's partitioned into a finite number od stable (resp. unstable) manifolds $W^\pm(C)=\{x\in X,\ \lim_{t\to\pm\infty}\phi_t(x)\in C\}$, where $\phi_t$ is the (global) flow of the gradient (or minus-gradient). Take a look to the following paper (pages 6-7):
The Atiyah-Guillemin-Sternberg convexity theorem
