Hypothetical negatively curved structures derived from graphite are described, in which all carbon atoms rest on triply periodic minimal surfaces (TPMS). The D minimal surface was calculated using the Weierstrass representation. By applying the Bonnet transformation to the D surface, the gyroid and P surfaces were constructed. Curvatures, densities, lattice parameters and energies have been calculated for all structures. The absolute value of the maximum Gaussian curvature is smaller than that for C60 fullerene. A new periodic graphite net with the same topology as the I-WP minimal surface, using 5-, 6- and 8-membered rings is found possible. The stability of 11 negatively curved graphitic structures has been determined using Tersoff's three-body potential. All the structures described are more stable than C60,mainly because the 120° bond angles in ordinary graphite are almost preserved in the 7- and 8-membered carbon rings. The way is now open to explore the decoration of minimal surfaces with further arrangements of atoms of different elements.