---
tag: A4E7SZ935M
kind: definition
name: Almgren–Pitts min-max theory
msc: ["58E12", 49Q05, 53A10]
defines: 6KVJHG1KRZ
uses: [7CBT4SM83Y, Z6QXZX1THM, 41X0YW2R9N, SSNFTHZ6YJ]
sources:
  - url: https://en.wikipedia.org/wiki/Almgren–Pitts_min-max_theory
---

## Definition

Let $(M, g)$ be a closed [[Riemannian manifold|7CBT4SM83Y]] of dimension $n + 1 \ge 3$.
Let $\mathcal{Z}_n(M; \Z_2)$ be the space of $n$-dimensional mod $2$ flat cycles in $M$ that are boundaries, with the flat topology.
These are the mod $2$ integral [[currents|Z6QXZX1THM]] $T = \partial U$, where $U$ is an $(n+1)$-dimensional mod $2$ integral current.
Let $\mathbf{M}(T)$ be the mass of $T$.

1. A *sweepout* of $M$ is a map $\Phi \colon [0, 1] \to \mathcal{Z}_n(M; \Z_2)$ with $\Phi(0) = \Phi(1) = 0$, continuous in the flat topology and with no concentration of mass.
It must represent the nontrivial class of $\pi_1(\mathcal{Z}_n(M; \Z_2), 0)$.
This group is $\Z_2$, since the Almgren isomorphism identifies it with $H_{n+1}(M; \Z_2)$.
2. The *width* of $M$ is
$$W(M) = \inf_{\Phi} \ \sup_{t \in [0, 1]} \mathbf{M}\big(\Phi(t)\big),$$
the infimum over all sweepouts.

*Almgren–Pitts min-max theory* is the theory of such min-max values, and of the analogous values over $k$-parameter families of cycles.
Its main theorem states that $W(M) > 0$.
Moreover, $W(M)$ is the area of a closed embedded [[minimal hypersurface|41X0YW2R9N]], counted with integer multiplicities.
This hypersurface is smooth when $3 \le n + 1 \le 7$.
In general its singular set has [[Hausdorff dimension|SSNFTHZ6YJ]] at most $n - 7$.

## Remarks

Pitts proved the smoothness for $3 \le n + 1 \le 6$, and the case $n + 1 = 7$ and the higher-dimensional statement follow from the regularity theory of Schoen and Simon.
The positivity of the width follows from an isoperimetric inequality: cycles of small mass bound regions of small volume, so a sweepout of small mass would be trivial.

Example: for the round unit sphere $S^3$, the level sets of a height function $x_4$ form a sweepout, $t \mapsto \partial\set{x_4 \le 2t - 1}$, whose largest member is the equatorial $2$-sphere, of area $4\pi$.
Here the width equals $4\pi$.

Non-example: the family $t \mapsto \partial B_{r(t)}(p)$ of boundaries of small geodesic balls, with $r(t) = \varepsilon \sin \pi t$ and $\varepsilon$ small, starts and ends at $0$ but is not a sweepout.
It is contractible in $\mathcal{Z}_n(M; \Z_2)$ relative to $0$, because the enclosed regions never fill $M$; so it represents the trivial class and fails clause 1.