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A non-negative expression, built from the norm of the 3-surface twistoroperator and the energy-momentum tensor of the matter fields on a spacelikehypersurface, is found which, in the asymptotically flat/hyperboloidal case,provides a lower bound for the ADM/Bondi--Sachs mass, while on closedhypersurfaces it gives the first eigenvalue of the Sen--Witten operator. Alsoin the closed case, its vanishing is equivalent to the existence of non-trivialsolutions of Witten's gauge condition. Moreover, it is vanishing if and only ifthe closed data set is in a flat spacetime with toroidal spatial topology. Thusit provides a positive definite measure of the strength of the gravitationalfield (with physical dimension mass) on closed hypersurfaces, i.e. some sort ofthe total mass of closed universes.

Mass, gauge conditions and spectral properties of the Sen-Witten and 3-surface twistor operators in closed universes