  In the first part of this paper we will work out a close and so far not yet
noticed correspondence between the swampland approach in quantum gravity and
geometric flow equations in general relativity, most notably the Ricci flow. We
conjecture that following the gradient flow towards a fixed point, which is at
infinite distance in the space of background metrics, is accompanied by an
infinite tower of states in quantum gravity. In case of the Ricci flow, this
conjecture is in accordance with the generalized distance and AdS distance
conjectures, which were recently discussed in the literature, but it should
also hold for more general background spaces. We argue that the entropy
functionals of gradient flows provide a useful definition of the generalized
distance in the space of background fields. In particular we give evidence that
for the Ricci flow the distance $\Delta$ can be defined in terms of the mean
scalar curvature of the manifold, $\Delta\sim\log \bar R$. For a more general
gradient flow, the distance functional also depends on the string coupling
constant.
  In the second part of the paper we will apply the generalized distance
conjecture to gravity theories with higher curvature interactions, like higher
derivative $R^2$ and $W^2$ terms. We will show that going to the weak coupling
limit of the higher derivative terms corresponds to the infinite distance limit
in metric space and hence this limit must be accompanied by an infinite tower
of light states. For the case of the $R^2$ or $W^2$ couplings, this limit
corresponds to the limit of a small cosmological constant or, respectively, to
a light additional spin-two field in gravity. In general we see that the limit
of small higher curvature couplings belongs to the swampland in quantum
gravity, just like the limit of a small $U(1)$ gauge coupling belongs to the
swampland as well.
