  We obtain estimates showing that on monotone symplectic manifolds
(asymptotic) spectral invariants of Hamiltonians which vanish on a non-empty
open set, U, descend to Ham_c(M\setminus U) from its universal cover.
Furthermore, we show these invariants and are continuous with respect to the
C^0-topology on Ham_c(M\setminus U). We apply these results to Hofer geometry
and establish unboundedness of the Hofer diameter of $Ham_c(M\setminus U)$ for
stably displaceable $U$. We also answer a question of F. Le Roux about
$C^0$-continuity properties of the Hofer metric.
