  The conformal invariance of the low energy limit theory governing the
electronic properties of graphene is explored. In particular, it is noted that
the massless Dirac theory in point enjoys local Weyl symmetry, a very large
symmetry. Exploiting this symmetry in the two spatial dimensions and in the
associated three dimensional spacetime, we find the geometric constraints that
correspond to specific shapes of the graphene sheet for which the electronic
density of states is the same as that for planar graphene, provided the
measurements are made in accordance to the inner reference frame of the
electronic system. These results rely on the (surprising) general
relativistic-like behavior of the graphene system arising from the combination
of its well known special relativistic-like behavior with the less explored
Weyl symmetry. Mathematical structures, such as the Virasoro algebra and the
Liouville equation, naturally arise in this three-dimensional context and can
be related to specific profiles of the graphene sheet. Speculations on possible
applications of three-dimensional gravity are also proposed.
