A variational characterization of calibrated submanifolds

TL;DR

This paper provides a variational characterization of calibrated submanifolds by showing they are critical points of volume under specific metric variations related to calibrations, extending previous results to various calibration types.

Contribution

It introduces a new variational framework for calibrated submanifolds without requiring the calibration to be closed, generalizing earlier work and including Cayley calibrations.

Findings

Calibrated submanifolds are critical points of volume under special metric variations.

The approach applies to almost complex, associative, coassociative, and Cayley calibrations.

The Cayley case exhibits unique behavior compared to other calibrations.

Abstract

Let $M$ be a fixed compact oriented embedded submanifold of a manifold $\overline{M}$. Consider the volume $\mathcal{V} (\overline{g}) = \int_M \mathsf{vol}_{(M, g)}$ as a functional of the ambient metric $\overline{g}$ on $\overline{M}$, where $g = \overline{g}|_M$. We show that $\overline{g}$ is a critical point of $\mathcal{V}$ with respect to a special class of variations of $\overline{g}$, obtained by varying a calibration $\mu$ on $\overline{M}$ in a particular way, if and only if $M$ is calibrated by $\mu$. We do not assume that the calibration is closed. We prove this for almost complex, associative, coassociative, and Cayley calibrations, generalizing earlier work of Arezzo-Sun in the almost K\"ahler case. The Cayley case turns out to be particularly interesting, as it behaves quite differently from the others. We also apply these results to obtain a variational characterization of Smith maps.