Free boundary minimal annuli in convex balls

Technical Summary: Free Boundary Minimal Annuli in Convex Balls

Problem Statement
This paper addresses two primary problems in the geometric analysis of free boundary minimal submanifolds within Riemannian 3-manifolds with boundary:

1. The 4-width of the Euclidean unit ball ($B^3$): While the first three widths of $B^3$ are known to be $\pi$ (realized by equatorial disks), the value of the fourth width, $\omega_4(B^3)$, remained open. Specifically, it was conjectured that $\omega_4(B^3)$ is realized by the area of the critical catenoid, the unique properly embedded free boundary minimal annulus in $B^3$ (up to congruence).
2. Multiplicity of free boundary minimal annuli: Motivated by Yau's conjecture on minimal spheres and White's conjecture on minimal tori in closed 3-spheres, the authors investigate the existence of multiple free boundary minimal annuli in compact 3-manifolds with nonnegative Ricci curvature and strictly convex boundary. While the existence of at least one such annulus was known, the existence of multiple (specifically three) with prescribed topology was an open problem.

Methodology
The authors employ the Almgren-Pitts min-max theory, specifically the Simon-Smith variant adapted for free boundary settings, combined with topological control techniques.

• Construction of Canonical Families: Inspired by the 5-parameter canonical family of Marques-Neves used to prove the Willmore conjecture on $S^3$, the authors construct a 4-parameter family of surfaces in $B^3$. This family is parameterized by $\mathbb{RP}^4$ and consists of properly embedded annuli (and their degenerations). The construction utilizes conformal diffeomorphisms of the ball and stereographic projections. A crucial step involves a "boundary blow-up" technique to ensure the continuity of the family as parameters approach the boundary of the parameter space, extending the family to a continuous map on $\mathbb{RP}^4$.
• Topological Control: The authors select a specific initial surface—the annulus obtained by stereographically projecting half of the Clifford torus from $S^3$. This choice is motivated by the "channel surface" structure of the Clifford torus, which is preserved under conformal deformations. This structure allows for rigorous topological control, ensuring that surfaces in the family have genus 0 and at most two boundary components (topological bound $(0, 1)$).
• Repetitive Min-Max Construction: To prove the existence of multiple annuli, the authors extend the 4-parameter family to a 6-parameter family (parameterized by $\mathbb{RP}^4 \times \mathbb{RP}^2$) by varying the rotational symmetry axis of the initial annulus. They then adapt the "repetitive min-max" scheme developed by Chu-Li for closed manifolds to the free boundary setting. This involves:
  • Iteratively applying the Simon-Smith min-max theorem to detect minimal surfaces.
  • Using a "relative pinch-off process" to remove detected minimal surfaces (disks or annuli) from the parameter space while maintaining the homotopy class of the sweepout.
  • Utilizing the Lyusternik-Schnirelmann theory on the cohomology of the parameter space to derive a lower bound on the number of critical points (minimal surfaces).

Key Contributions and Results

1. Upper Bound on $\omega_4(B^3)$:
   The authors prove that the Simon-Smith min-max width of their constructed 4-parameter family is achieved by a multiplicity-one properly embedded free boundary minimal annulus. By combining the topological control (which rules out disks) and Morse index estimates (showing the index is exactly 4), they identify this limit surface as the critical catenoid. Consequently, they establish the upper bound:
   $$\omega_4(B^3) \le \text{Area}(\text{Critical Catenoid})$$
   Combined with a lower bound of $\pi$ (proven via a gap theorem for free boundary minimal surfaces), this yields:
   $$\pi < \omega_4(B^3) \le \text{Area}(\text{Critical Catenoid})$$

2. Existence of Three Minimal Annuli:
   The paper proves that every compact 3-manifold with nonnegative Ricci curvature and strictly convex boundary contains at least three embedded free boundary minimal annuli. This is achieved by:

   • Constructing the 6-parameter family $\Phi_6$.
   • Applying the repetitive min-max process to detect a sequence of minimal surfaces.
   • Using a cohomological argument involving the non-vanishing of $\lambda^4 \cup \alpha^2$ in the cohomology ring of the parameter space $Y = \mathbb{RP}^4 \times \mathbb{RP}^2$.
   • Showing that if fewer than three annuli existed, the topological constraints would force a contradiction with the non-vanishing cohomology class, implying the existence of at least three distinct minimal annuli.

3. Multiplicity One and Topological Control:
   The authors rigorously establish the Multiplicity One Theorem for free boundary Simon-Smith min-max limits in this context, ensuring that the detected surfaces are not multiple covers. They also provide a framework for controlling the topology (genus and boundary complexity) of min-max surfaces in the free boundary setting, adapting results from Franz-Schulz and Ketover.

Significance
The paper claims significance in several areas:

• Free Boundary Analog of Willmore Conjecture: It provides the first natural 4-parameter family of surfaces whose min-max width is realized by the critical catenoid, serving as a free boundary analog to the Marques-Neves construction for the Willmore conjecture.
• Resolution of Width Conjecture: It confirms the conjecture that the 4-width of the unit ball is bounded above by the area of the critical catenoid, narrowing the gap for the exact value of $\omega_4(B^3)$.
• Multiplicity Results: It represents the first construction of multiple free boundary minimal surfaces with prescribed topology (specifically, genus 0 with two boundary components) in general convex 3-manifolds. This extends previous results on minimal disks and tori in closed manifolds to the free boundary setting.
• Methodological Advancement: The development of the "relative pinch-off process" and the adaptation of repetitive min-max theory to free boundary settings provide a new toolkit for constructing minimal submanifolds with specific topological types in manifolds with boundary.

The authors note that while they establish the upper bound for $\omega_4(B^3)$, the strict equality $\omega_4(B^3) = \text{Area}(\text{Critical Catenoid})$ remains conditional on the uniqueness of index-4 free boundary minimal surfaces or a lower bound on their area, which are currently open questions. Similarly, the existence of exactly three annuli is proven under the assumption of finiteness, which is justified by compactness arguments for the specific topological class.