  Carbon atoms can form structures in one, two, and three dimensions due to its
unique chemical versatility. In terms of thermal conductivity, carbon
polymorphs cover a wide range from very low values with amorphous carbon to
very high values with diamond, carbon nanotubes and graphene. Schwarzites are a
class of three-dimensional fully covalent sp$^2$-bonded carbon polymorphs, with
the same local chemical environment as graphene and carbon nanotubes, but
negative Gaussian curvature. We calculate the thermal conductivity of a (10,0)
carbon nanotube, graphene, and two schwarzites with different curvature, by
molecular dynamics simulations based on the Tersoff empirical potential. We
find that schwarzites present a thermal conductivity two orders of magnitude
smaller than nanotubes and graphene. The reason for such large difference is
explained by anharmonic lattice dynamics calculations, which show that phonon
group velocities and mean free paths are much smaller in schwarzites than in
nanotubes and graphene. Their reduced thermal conductivity, in addition to
tunable electronic properties, indicate that schwarzites could pave the way
towards all-carbon thermoelectric technology with high conversion efficiency.
