  We prove rigidity results involving the Hawking mass for surfaces immersed in
a $3$-dimensional, complete Riemannian manifold $(M,g)$ with non-negative
scalar curvature (resp. with scalar curvature bounded below by $-6$). Roughly,
the main result states that if an open subset $\Omega\subset M$ satisfies that
every point has a neighbourhood $U\subset \Omega$ such that the supremum of the
Hawking mass of surfaces contained in $U$ is non-positive, then $\Omega$ is
locally isometric to Euclidean ${\mathbb R}^3$ (resp. locally isometric to the
Hyperbolic 3-space ${\mathbb H}^3$). Under mild asymptotic conditions on the
manifold $(M,g)$ (which encompass as special cases the standard "asymptotically
flat" or, respectively, "asymptotically hyperbolic" assumptions) the previous
quasi-local rigidity statement implies a \emph{global rigidity}: if every point
in $M$ has a neighbourhood $U$ such that the supremum of the Hawking mass of
surfaces contained in $U$ is non-positive, then $(M,g)$ is globally isometric
to Euclidean ${\mathbb R}^3$ (resp. globally isometric to the Hyperbolic
3-space ${\mathbb H}^3$). Also, if the space is not flat (resp. not of constant
sectional curvature $-1$), the methods give a small yet explicit and strictly
positive lower bound on the Hawking mass of suitable spherical surfaces. We
infer a small yet explicit and strictly positive lower bound on the Bartnik
mass of open sets (non-locally isometric to Euclidean ${\mathbb R}^{3}$) in
terms of curvature tensors. Inspired by these results, in the appendix we
propose a notion of "sup-Hawking mass" which satisfies some natural properties
of a quasi-local mass.
