  We consider gapped fractional quantum Hall states on the lowest Landau level
when the Coulomb energy is much smaller than the cyclotron energy. We introduce
two spectral densities, \rho_T(\omega) and \bar \rho_T(\omega), which are
proportional to the probabilities of absorption of circularly polarized
gravitons by the quantum Hall system. We prove three sum rules relating these
spectral densities with the shift S, the q^4 coefficient of the static
structure factor S_4, and the high-frequency shear modulus of the ground state
\mu_\infty, which is precisely defined. We confirm an inequality, first
suggested by Haldane, that S_4 is bounded from below by |S-1|/8. The Laughlin
wavefunction saturates this bound, which we argue to imply that systems with
ground state wavefunctions close to Laughlin's absorb gravitons of
predominantly one circular polarization. We consider a nonlinear model where
the sum rules are saturated by a single magneto-roton mode. In this model, the
magneto-roton arises from the mixing between oscillations of an internal metric
and the hydrodynamic motion. Implications for experiments are briefly
discussed.
