2015/11/30 by Yongsheng Zhang
Mathematics · #math.DG #math.AT #msc:53C38 #msc:49Q15 #msc:28A75
published as Advances in Mathematics, Volume 308, 21 February 2017, Pages 645-670 · Improved Version. arXiv admin note: text overlap with arXiv:1501.01836
arxiv created 2019/06/25 · arxiv updated 2019/06/27
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 §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.