Almgren–Pitts min-max theory

Definition

Let $(M, g)$ be a closed Riemannian manifold 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 $T = \partial U$, 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, counted with integer multiplicities. This hypersurface is smooth when $3 \le n + 1 \le 7$. In general its singular set has Hausdorff dimension at most $n - 7$.