We use ab initio density functional calculations to study the stability, elastic properties and electronic structure of sp2 carbon minimal surfaces with negative Gaussian curvature, called schwarzites. We focus on two systems with cubic unit cells containing 152 and 200 carbon atoms, which are metallic and very rigid. The porous schwarzite structure allows for efficient and reversible doping by electron donors and acceptors, making it a promising candidate for the next generation of alkali ion batteries. We identify schwarzite structures that act as arrays of interconnected spin quantum dots or become magnetic when doped. We introduce two interpenetrating schwarzite structures that may find their use as the ultimate super-capacitor.