Every finite graph arises as the singular set of a compact 3‐D calibrated area minimizing surface

Abstract Given any (not necessarily connected) combinatorial finite graph and any compact smooth 6‐manifold with the third Betti number , we construct a calibrated 3‐dimensional homologically area minimizing surface on equipped in a smooth metric , so that the singular set of the surface is precisely an embedding of this finite graph. Moreover, the calibration form near the singular set is a smoothly twisted special Lagrangian form. The constructions are based on some unpublished ideas of Professor Camillo De Lellis and Professor Robert Bryant.