On extending calibration pairs

Abstract

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 $\mathbb R$-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 {\S}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.