  Temperature constraints are highly desirable in the experimental setup when
seeking the synthesis of new carbon structures. Fluctuations of the Dirac field
result in temperature-dependent corrections to the Helfrich-Canham formulation,
which governs the classical elasticity of the membrane in equilibrium state.
Here, we examine the emergent shapes allowed by the effective model up to
quadratic order in Ricci curvature and discuss the constraints required to
observe them. We determine the mechanical stability conditions and provide a
phase diagram characterized by the appearance of a critical temperature $T_c$
that distinguishes between carbon nanotube and fullerene phases. The
observation of minimal and developable surfaces is anticipated, respectively,
at the high- and low-temperature regimes.
