Flat or crumpled: states of active symmetric membranes

We set up and study the hydrodynamic theory for inversion-symmetric active fluid and tethered membranes. For some choices of the activity parameter, such membranes are stable and described by linear hydrodynamic equations, which are exact in the asymptotic long wavelength limit, giving stable flat phases with translational quasi long range orders. For other choices of the activity parameter, the system is linearly unstable in the long wavelength limit, implying crumpling, or has intermediate wavevector instabilities, suggesting patterns. We argue that in such an active membrane thermal noises dominate over any active noises, and use those to calculate the correlation functions of membrane conformation fluctuations in the stable case, and the associated correlation functions of the embedding bulk flow velocities