On extending calibration pairs

TL;DR

This paper explores how to extend local calibration pairs to global ones in Riemannian manifolds, revealing new mass-minimizing properties and conditions under which submanifolds can or cannot be calibrated globally.

Contribution

It introduces methods for extending calibration pairs to global settings and demonstrates the existence of mass-minimizing submanifolds that cannot be globally calibrated.

Findings

Homologically nontrivial submanifolds can be mass-minimizing under certain conformal metrics.

Extension theorems for calibration pairs are established for singular and multi-component submanifolds.

Existence of mass-minimizing submanifolds without smooth calibrations in some manifolds.

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.