Min-max theory and minimal surfaces with prescribed genus

TL;DR

The paper develops a topological min-max framework to produce embedded minimal surfaces of prescribed genus $g$ in closed 3-manifolds with positive Ricci curvature, starting from Simon--Smith families of genus at most $g$. The core method combines repetitive min-max to generate a topologically optimal family with controlled area, together with pinch-off interpolations that continuously lower genus across a family while preserving key geometric bounds. A central result shows that if a Simon--Smith family detects nontrivial relative topology between surface families of genus $\le g$ and $\le g-1$, then an orientable embedded minimal surface of genus $g$ exists with area bounded by the min-max width $L$. The framework relies on Frankel-type properties, multiplicity-one results for minimal surfaces, and a detailed construction of deformation via pinch-off processes, enabling applications toward constructing multiple minimal surfaces of prescribed genus in Ricci-positive 3-spheres. Collectively, the results provide a robust topological route to controlled-genus minimal surfaces with quantitative area bounds in positively curved 3-manifolds.

Abstract

We establish a general min-max type theorem that produces minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature. An important intermediate step is to show that, in a generic metric with positive Ricci curvature, any family of smooth embedded surfaces, possibly with finitely many singularities, can be deformed into a certain ``topologically optimal family". Results in this paper will be crucial to our program on the construction of multiple minimal surfaces with prescribed genus in 3-spheres via topological methods.

Figures (7)

• Figure 1: This schematic diagram shows the set $\mathcal{S}_{\leq g}(M)$ of all surfaces with genus at most $g$, possibly with singularities. The gray cap below represents $\mathcal{S}_{\leq g-1}(M)$, while the blue path represents the image of a map $\Phi:X\to\mathcal{S}_{\leq g}(M)$, detecting some non-trivial relative structure of the pair $(\mathcal{S}_{\leq g}(M),\mathcal{S}_{\leq g-1}(M))$.
• Figure 2: This is an example of pinch-off process. It has one neck-pinch surgery, and one connected component shrunk to a point.
• Figure 3: This picture shows an element $S\in\mathcal{S}(M)$: It is a closed set, which contains a (black) smooth surface part and also the red points. The smallest possible punctate set for $S$ is given by the red points. Note $S$ has genus $1$.
• Figure 4: In the first picture, the red region is $X^{K-1}$, the blue is $E^{K-1}$, and the green is $E^{K-2}$. In the second picture, the gray surface (with suitable smoothening) denotes $\Xi^{K-1}$. In the last picture, the gray surface denotes $\Xi^{K-2}$.
• Figure 5: Constructing $\widetilde{c}$ from $c$

Theorems & Definitions (72)

• Theorem 1.2
• Remark 1.3
• Theorem 1.4: Existence of a topologically optimal family
• Definition 2.1: Punctate surface
• Definition 2.2: Punctate set
• Remark 2.3
• Definition 2.4
• Definition 2.5: Genus of a punctate surface
• Lemma 2.6
• Definition 2.7: Simon--Smith family