Enumerative Min-Max Theorem

• Enumerative Min-Max Theorem is a framework that links algebraic-topological invariants of parameter spaces to the existence and count of embedded minimal surfaces.
• It employs p-sweepouts, min-max widths, and homological non-deformability conditions to ensure that critical varifolds arise as multiplicity one, two-sided minimal hypersurfaces.
• Applications include explicit lower bounds on minimal surface counts in Ricci-positive 3-manifolds and asymptotic analysis of p-widths via geometric measure theory.

The enumerative min-max theorem establishes foundational links between the topology of spaces of surfaces, specifically of prescribed genus, and the existence and enumeration of minimal surfaces via min-max variational techniques in geometric analysis. It unifies the algebraic-topological invariants of parameter spaces with the multiplicity and distinction of minimal hypersurfaces produced by min-max theory. Rigorous formulations by Chu–Li, Chu–Li–Wang, and others underpin modern enumerative minimal surface theory in Riemannian 3-manifolds, particularly those with positive Ricci curvature, culminating in explicit lower bounds on the number of distinct embedded minimal surfaces of fixed genus and multiplicity one (Chu et al., 2023, Chu et al., 5 Jan 2026).

1. Min-Max Theory and $p$-Widths

Let $(M^{n+1}, g)$ be a closed Riemannian manifold with $3 \leq n+1 \leq 7$. The space $\mathcal Z_n(M;\mathbb Z_2)$ of flat $n$-cycles with $\mathbb Z_2$ coefficients, endowed with the flat topology, serves as the primary parameter space for min-max constructions. The Almgren–Pitts theory asserts this space is (weakly) homotopy equivalent to $\mathbb{R}P^\infty$, supporting a single nonzero cohomology generator $\bar\lambda \in H^1(\mathcal Z_n; \mathbb Z_2)$. For a finite simplicial complex $X$ and a continuous map

$\Phi: X \to \mathcal Z_n(M; \mathbb Z_2)$

constituting a $p$-sweepout, the min-max width is defined as

$\omega_p(M, g) = \inf_{\substack{\Phi\ \textrm{%%%%10%%%%–sweepout}}} \sup_{x\in X} \mathbf M_g(\Phi(x)),$

where $$\Phi^*(\bar\lambda)^p \neq 0 \in H^p(X; \mathbb Z_2)$$ ensures topological nontriviality. The critical set associated to a minimizing sequence $\{ \Phi_i \}$ consists of varifold limits at level $\omega_p$, encoding the occurrence of minimal hypersurfaces.

2. Strong Multiplicity One and Critical Set Structure

The strong multiplicity one theorem (Chu–Li) asserts that for any closed $(M^{n+1}, g)$ ($3 \leq n+1 \leq 7$) with bumpy metric or $\Ric_g>0$, and for every $p$, there exists a pulled-tight minimizing sequence of $p$-sweepouts for width $\omega_p(M, g)$ such that every varifold in the corresponding critical set arises from a multiplicity one, embedded, two-sided minimal hypersurface. The theorem employs a homological version of min-max restricted to upper mass bounds and performs $(\varepsilon, \delta)$-deformations to eliminate higher multiplicity and one-sided cycles from the critical set, ultimately yielding only “good” (i.e., multiplicity one, two-sided) minimal hypersurfaces in the limit (Chu et al., 2023).

Key technical components include:

• Pitts’ notion of $(m, r)$–almost-minimizing varifold and annular replacements;
• An $(\varepsilon, \delta)$-deformation lemma following Marques–Neves;
• Homological min-max schemes with boundary and upper mass constraint.

3. Enumerative Min-Max Theorem: Topological and Homological Input

Chu–Li–Wang formulated a precise enumerative min-max theorem for minimal surfaces of fixed genus in closed, orientable, Ricci-positive 3-manifolds (Chu et al., 5 Jan 2026). For the space $\mathcal S(M)$ of punctate surfaces (finite-area $2$-dimensional sets, smooth except for finitely many points), $\mathcal S_{\leq g}(M)$ denotes those of genus $\leq g$. A Simon–Smith family

$$\Phi : (X, Z) \to (\mathcal S_{\leq g}(M), \mathcal S_{\leq g-1}(M))$$

parametrizes a sweepout by genus $\leq g$ surfaces, with boundary landing in genus $\leq g-1$.

Given a relative homology class $w \in H_k(X, Z; \mathbb Z/2)$ and $p$ cohomology classes $\lambda_i \in H^{k_i}(X\setminus Z;\mathbb Z/2)$, let $$\omega=\lambda_1\smile \cdots \smile \lambda_p \in H^{s}(X\setminus Z)$$ ($s=k_1+\dots+k_p$). The construction enforces two conditions:

1. Non-deformability in homology: For any $(Y, \partial Y)\subset(X, Z)$ representing $w\frown\omega$, the restricted family cannot be homotoped by pinch-off processes into $\mathcal S_{\leq g-1}(M)$.
2. Avoidance of trivial index bounds: For each $i$, whenever a $k_i$-cycle with nonzero $\lambda_i$ is present, the restriction cannot be deformed near a single smooth genus $g$ surface.

The theorem asserts that

$\#\bigl\{ \text{embedded minimal surfaces in }(M, g)\text{ of genus }g\text{ and area}\leq \max_{x \in X}\Area(\Phi(x)) \bigr\} \geq p+1.$

This lower bound is realized by producing at least $p+1$ minimal surfaces of genus $g$.

4. Applications: Explicit Surface Counts and Generalizations

A notable application is the explicit construction of at least four embedded minimal surfaces of genus $2$ in any 3-sphere of positive Ricci curvature (Chu et al., 5 Jan 2026). Using a $13$-parameter Simon–Smith family, built from polynomial sweepouts and group actions, and verifying the presence of three nonzero cohomology classes $\alpha \in H^1(X; \mathbb Z/2)$ with $\alpha^3 \neq 0$, the lower bound is achieved by applying the enumerative min-max theorem with $p=3$. The proof method extends to higher genus surfaces and other manifolds, conditioned on the existence of sweepout parameter spaces $X$ with sufficient topological complexity as encoded in the relevant cohomology.

Generalizations rely on producing appropriate parameter families and verifying the technical non-deformability and avoidance conditions, as well as regularity through arguments such as those by Wang–Zhou for multiplicity one and Simon–Smith for minimizing two-sidedness.

5. Corollaries and Asymptotics in Enumerative Theory

From the strong multiplicity one theorem and its enumerative consequences, several corollaries arise:

• The $p$-widths $\omega_p(M, g)$ form a strictly increasing sequence, and each is realized by an embedded, two-sided minimal hypersurface with area $\omega_p$ and Morse index at most $p$, yielding infinitely many distinct such hypersurfaces (Chu et al., 2023).
• The Weyl law describes the asymptotic growth:

$\omega_p(M, g) \sim c_n\, p^{1/(n+1)}$

as $p\to\infty$, ensuring that the areas of these surfaces diverge and confirming their distinction (Chu et al., 2023).

• In positive Ricci curvature or for bumpy metrics, surfaces are necessarily separating and have multiplicity exactly one as $\mathbb Z_2$ cycles.
• In the genus enumeration setting, parameter space topological invariants (cup and cap products) control the minimal number of genus $g$ surfaces.

Theorem/Result	Hypotheses	Consequence
Strong multiplicity one (Chu et al., 2023)	Closed $M^{n+1},\, 3\leq n+1\leq 7$, bumpy or $\Ric>0$	Critical varifolds are all multiplicity one
Enumerative min-max (Chu et al., 5 Jan 2026)	Ricci-positive $3$-manifold, genus $g$ sweepout	At least $p+1$ genus $g$ minimal surfaces

6. Context, Implications, and Open Directions

The enumerative min-max theorem merges Lyusternik–Schnirelmann–type invariants from algebraic topology with geometric PDE approaches, concretely relating the topology of sweepout-spaces (e.g. $\mathcal S_{\leq g}(M)$) to counts of minimal surfaces. The program, initiated in previous work and culminating in (Chu et al., 5 Jan 2026), establishes a template for producing explicit lower bounds for embedded minimal surfaces of prescribed genus via parameter space topology.

Open problems include:

• Determining the exact minimal number $\mathfrak n_g(S^3)$ of genus $g$ minimal surfaces in Ricci-positive $S^3$ (conjecturally $\mathfrak n_1(S^3)=5$, $\mathfrak n_2(S^3)=4$).
• Extension to non-Ricci-positive and higher-dimensional manifolds.
• Analysis of the interaction between the homotopical complexity of $\mathcal S_{\leq g}(M)$ and the variational min-max width sequence.
• Investigation of bifurcation phenomena and degeneration near special metrics.

The enumerative min-max theorem thus offers a paradigm for quantifying and classifying minimal surfaces via a blend of geometric measure theory, variational analysis, and algebraic topology, with broad ramifications for both the analytical and topological understanding of minimal hypersurfaces (Chu et al., 5 Jan 2026, Chu et al., 2023).