An enumerative min-max theorem for minimal surfaces

TL;DR

The paper introduces a robust topological (Simon–Smith) min-max framework to enumerate minimal surfaces of prescribed genus in closed 3-manifolds with positive Ricci curvature. By combining a perturbation to generic metrics, a topologically optimal deformation, and a careful cap-product analysis, it shows that suitable cohomological data force the existence of at least $p+1$ orientable genus-$g$ minimal surfaces with area bounded by the subfamily’s maximal area. The authors construct explicit high-dimensional parameter families, notably a 7-parameter family $\Psi$ and a 13-parameter family $\Xi$ with $D_{24}$ symmetry, in the setting of $S^3$, to conclude the existence of four embedded genus-$2$ minimal surfaces in any positive Ricci curvature 3-sphere. The work develops a detailed topological analysis of the parameter space and consumes a sequence of algebraic-topological lemmas to validate the min-max hypotheses, thereby enabling a concrete genus-2 lower bound and illuminating a path toward higher-genus results. The approach provides a versatile, black-box framework for enumerating minimal surfaces of fixed genus via purely topological data, with potential implications for understanding the landscape of minimal submanifolds in Ricci-positive 3-manifolds.

Abstract

We prove an enumerative min-max theorem that relates the number of genus g minimal surfaces in 3-manifolds of positive Ricci curvature to topological properties of the set of embedded surfaces of genus $\leq g$, possibly with finitely many singularities. This completes a central component of our program of using topological methods to enumerating minimal surfaces with prescribed genus. As an application, we show that every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.

Figures (3)

• Figure 1: On the left is an example where $X$ is the annulus, $Z$ is the two boundary circles, $Y$ is the blue path, $w$ is the fundamental class $[X]$, $p=1$, and $\lambda_1$ is the Poincaré dual of $[Y]$, meaning $[Y]=[X]\frown \lambda_1$. Note the pairing of $\lambda_1$ with the red loop is 1. On the right, we have a schematic where the graphical surface represents $\mathcal{S}_{\leq g}(M)$, with the two boundary circles being $\mathcal{S}_{\leq g-1}(M)$. The blue path is the family $\Phi|_Y$, and the two red critical points are the minimal surfaces we are searching for.
• Figure 2: An example of the surface $\Sigma$: It shows a portion of a surface $\Sigma$ near the circle $\{x_1=-b_2,x_2=-b_1\}\subset S^3$, which is represented by the $z$-axis in the figure, parametrized by $\alpha$. Those $\alpha$ satisfying $F (\mathbf{x}(\alpha),\alpha)=0$ correspond to the points on the surface with horizontal tangent planes.
• Figure 3: The cylinder, after suitable boundary identification, is $S^3$. Then the red pieces form a loop, and so do the blue pieces. Together they form a link of the form $\Upsilon'_\circ(Z;\mu^+,\mu^-).$

Theorems & Definitions (88)

• Theorem 1.1: Enumerative min-max theorem
• Remark 1.2
• Theorem 1.3
• Definition 2.1: Punctate surface
• Definition 2.2: Simon--Smith family
• Theorem 2.3: Existence of a topologically optimal family
• Proposition 2.4
• Lemma 3.1
• proof
• Proposition 3.2