BY analogy with the positively curved carbon networks that comprise the fullerenes 1-3, it has been suggested 4 that negative curvature might be possible in graphitic carbon sheets, giving rise to extended structures corresponding to periodic minimal surfaces 5 that divide space into two disjoint labyrinths. Whereas the positive curvature of fullerenes results from the presence of five-membered rings, negative curvature would derive from seven-membered rings. Here we present calculations of the cohesive energy and bulk moduli of two such hypothetical, negatively curved carbon networks. We find that both have a cohesive energy smaller than that of graphite but significantly greater than that of C60, even though the proportion of odd-membered rings is comparable. We therefore suggest that it is worth scrutinizing the insoluble residue generated in the carbon-arc preparation of fullerenes 6 for possible evidence of fragments of negatively curved graphitic carbon.