  It is well known that viable architectural structures can be identified by
locating the critical points of the gravitational potential energy congruent
with some fixed surface metric. This is because, if the walls are thin, the
lowest energy modes of deformation are strain-free, and thus described by
surface isometries. If it is to stand, however, an arch had better possess some
minimum rigidity. The bending energy consistent with this construction
protocol, we will show, can only depend on curvature deviations away from the
reference equilibrium form. The question of stability, like the determination
of equilibrium, turns on the geometry. We show how to construct the
self-adjoint operator controlling the response to deformations consistent with
isometry. As illustration, we reassess the stability of a simple catenary arch
in terms of the behavior of the ground state of this operator. The energy of
this state increases monotonically with the bending rigidity and it is possible
to identify the critical rigidity above which the arch is rendered stable.
While this dependence may be monotonic, it exhibits a number of subcritical
kinks indicating significant qualitative changes in the ground state associated
with eigenvalue crossovers among the unstable modes in the spectrum; the number
of such competing modes increasing rapidly as the rigidity is lowered. The
initial collapse of a subcritical arch is controlled by the ground state; on
the critical threshold, there are two unstable modes of equal energy, one
raising the arch at its center, the other lowering it. The latter dominates as
the instability grows. The qualitative behavior of the ground state changes as
the rigidity is lowered---its nodal pattern as well as its parity undergoing
abrupt changes as the intervals between crossovers converge---complicating the
prediction of the dominant initial mode of collapse.
