Let $T=\mathbb{C}^{*}$ act on a smooth complex *quasi*-projective variety $X$. Assume that the limit point $\lim_{t\to 0}t\cdot x$ exists for every $x\in X$.

From the induced $U(1)$-action and its (real) moment map one can cook up a Morse function (or, if the fixed points are not isolated, a Morse-Bott function) $f$ on $X$, whose gradient flow $\varphi_s$ is equal to $\varphi_s (x)=e^{s}\cdot x$ for $s\in\mathbb{R}$ (considering $e^s\in T$). So, in particular, the limit of $s\cdot x$ for $s\to -\infty$ exists for every $x\in X$.

**Edit:** Let $X^{-}$ be the subset of those $x\in X$ for which
$$\lim_{t\to \infty} t\cdot x$$
exists. Assume that $X^-$ is a compact subset of $X$.

Q.Under the above hypotheses, when are the sub-level sets $f^{-1}\left((-\infty,c]\right)$ of $f$ compact for every $c\in\mathbb{R}$? Any references about this?