---
tag: KRK48SX454
kind: definition
name: Area-minimizing current
msc: [49Q15, 49Q05, 49Q20]
defines: 4NKP73KVVW
uses: [EG4AJVEFT2, 360H7MD9RD, 0E3G0GRPZ6, 0AKRKVHW94]
sources:
  - url: https://en.wikipedia.org/wiki/Current_(mathematics)
---

## Definition

Let $1 \le m \le n$, and let $T$ be an [[integral current|EG4AJVEFT2]] of dimension $m$ on $\R^n$, with mass $\mathbf{M}(T)$ and boundary $\partial T$.
The current $T$ is *area-minimizing*, or *mass-minimizing*, if
$$\mathbf{M}(T) \le \mathbf{M}(S)$$
for every integral $m$-current $S$ on $\R^n$ with $\partial S = \partial T$.

## Remarks

A local version is used for currents in an open set $U$.
There one compares $T$ with $S$ only when $T - S$ is the boundary of an integral current with compact support in $U$.
For a compact oriented $C^1$ submanifold, the mass of the associated [[current|360H7MD9RD]] is its area.
So an area-minimizing submanifold has least area among all oriented competitors with the same boundary, smooth or not.

Example: let $D$ be the closed unit disc in the plane $x_3 = 0$ of $\R^3$, oriented by $e_1 \wedge e_2$, and let $\omega = dx_1 \wedge dx_2$, a closed [[differential form|0E3G0GRPZ6]] of comass $1$.
For every integral $2$-current $S$ with $\partial S = \partial [\![D]\!]$, one has $S - [\![D]\!] = \partial R$ for an integral $3$-current $R$, since every integral cycle in $\R^3$ bounds.
Then $S(\omega) - [\![D]\!](\omega) = R(d\omega) = 0$ by the definition of the boundary, as in [[Stokes' theorem|0AKRKVHW94]].
So $\mathbf{M}(S) \ge S(\omega) = [\![D]\!](\omega) = \pi = \mathbf{M}([\![D]\!])$, and $[\![D]\!]$ is area-minimizing.

Non-example: the upper unit hemisphere, oriented so that its boundary is $\partial [\![D]\!]$, has mass $2\pi > \pi$, so it is not area-minimizing.

## Explanations

### L1 human

A current is area-minimizing if no other integral current with the same boundary has smaller mass.