  We demonstrate the irreversibility of a wide class of world-sheet
renormalization group (RG) flows to first order in $\alpha'$ in string theory.
Our techniques draw on the mathematics of Ricci flows, adapted to
asymptotically flat target manifolds. In the case of somewhere-negative scalar
curvature (of the target space), we give a proof by constructing an entropy
that increases monotonically along the flow, based on Perelman's Ricci flow
entropy. One consequence is the absence of periodic solutions, and we are able
to give a second, direct proof of this. If the scalar curvature is everywhere
positive, we instead construct a regularized volume to provide an entropy for
the flow. Our results are, in a sense, the analogue of Zamolodchikov's
$c$-theorem for world-sheet RG flows on noncompact spacetimes (though our
entropy is not the Zamolodchikov $C$-function).
