On extending calibration pairs

The paper studies how to extend local calibration pairs to global ones in various situations. As a result, new discoveries involving mass-minimizing properties are exhibited. In particular, we show that a R-homologically nontrivial connected submanifold M of a smooth Riemannian manifold X is homologically mass-minimizing for some metrics in the same conformal class. Moreover, several generalizations for M with multiple connected components or for a mutually disjoint collection (see §3.5) are obtained. For a submanifold with certain singularities, we also establish an extension theorem for generating global calibration pairs. By combining these results, we find that, in some Riemannian manifolds, there are homologically mass-minimizing smooth submanifolds which cannot be calibrated by any smooth calibration.