Cheeger bounds on spin-two fields

We consider gravity compactifications whose internal space consists of small
bridges connecting larger manifolds, possibly noncompact. We prove that, under
rather general assumptions, this leads to a massive spin-two field with very
small mass. The argument involves a recently-noticed relation to Bakry--\'Emery
geometry, a version of the so-called Cheeger constant, and the theory of
synthetic Ricci lower bounds. The latter technique allows generalizations to
non-smooth spaces such as those with D-brane singularities. For AdS$_d$ vacua
with a bridge admitting an AdS$_{d+1}$ interpretation, the holographic dual is
a CFT$_d$ with two CFT$_{d-1}$ boundaries. The ratio of their degrees of
freedom gives the graviton mass, generalizing results obtained by Bachas and
Lavdas for $d=4$. We also prove new bounds on the higher eigenvalues. These are
in agreement with the spin-two swampland conjecture in the regime where the
background is scale-separated; in the opposite regime we provide examples where
they are in naive tension with it.