We have studied the stability, elastic properties and electronic structure of sp² carbon minimal surfaces with negative Gaussian curvature, called schwarzites. We focused 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 electrondonors and acceptors, amking it a promising candidate for the next generation of alkali ion batteries. We identified schwarzite structures that act as arrays of interconnected spin quantum dots or become magnetic when doped. We introduced two interpenetrating scwarzite structures that may find their use as the ultimate super-capacitor.