Reentrant Rigidity Transition in Planar Epithelia with Volume and Area Elasticity

We find a reentrant columnar-to-squamous rigidity transition in three-dimensional (3D) epithelia, governed by volume and area elasticity. Our model maps to the classic 2D area- and perimeter-elasticity model but, unlike its 2D counterpart, shows compression-induced softening or stiffening, depending on the initial state. The phase diagram reveals floppy states with vanishing shear and in-plane bulk moduli, alongside a lateral-tension-driven discontinuous columnar-to-squamous transition. The critical behavior underlying the emergence of the reentrant transition belongs to the mean-field universality class.