A bound on the free energy of tensionless membranes

Using the proof of Willmore's conjecture by Marques and Neves, we conjecture that the free energy of tensionless fluid membranes of arbitrary genus has an upper bound. This implies that the average genus of such a membrane, in equilibrium, is finite, regardless of external constraints. We propose that the Gaussian rigidity may be determined by measuring the relative frequencies of large-genus configurations at low temperature.