---
abstract: 'We provide an introduction to the theory of calibrated submanifolds through the key examples related with special holonomy. We focus on calibrated geometry in Calabi–Yau, $\operatorname{G}_2$ and $\operatorname{Spin}(7)$ manifolds, and describe fundamental results and techniques in the field.'
author:
- |
    Jason D. Lotay\
    [University College London]{}
title: Calibrated Submanifolds
---

Introduction {#introduction .unnumbered}
============

A key aspect of mathematics is the study of variational problems. These can vary from the purely analytic to the very geometric. A classic geometric example is the study of geodesics, which are critical points for the length functional on curves. As we know, understanding the geodesics of a given Riemannian manifold allows us to understand some of the ambient geometry, for example the curvature. The higher dimensional analogue would be to study critical points for the volume functional, and we would hope (and it indeed turns out to be the case) that these critical points, called *minimal submanifolds*, encode crucial aspects of the geometry of the manifold.

Just like the geodesic equation, we would expect (and it is true) that minimal submanifolds are defined by a (nonlinear) second order partial differential equation. Such equations are very difficult to solve in general, so 