*Accepted Paper*

**Inserted:** 27 dec 2012

**Last Updated:** 27 feb 2014

**Journal:** Comm. Math. Phys.

**Year:** 2013

**Abstract:**

We investigate ground state configurations for atomic potentials including both two- and three-body nearest-neighbor interaction terms. The aim is to prove that such potentials may describe crystallization in carbon nanostructures such as graphene, nanotubes, and fullerenes. We give conditions in order to prove that planar energy minimizers are necessarily honeycomb, namely graphene patches. Moreover, we provide an explicit formula for the ground state energy which exactly quantifies the lower-order surface energy contribution. This allows to give some description of the geometry of ground states. By recasting the minimization problem in three-space dimensions, we prove that ground states are necessarily nonplanar and, in particular, rolled-up structures are energetically favorable. Eventually, we check that the C$_{20} $ and C$_{60}$ fullerenes are strict local minimizers, hence stable.

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