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Existence of free boundary minimal disks in convex regions

The paper establishes the existence of at least one embedded free boundary minimal disk in any mean convex three-ball, and proves the existence of at least three such disks when the domain is strictly convex with nonnegative Ricci curvature, utilizing a multiplicity-one theorem for free boundary Simon-Smith min-max theory.

Original authors: Lorenzo Sarnataro, Douglas Stryker, Zhichao Wang, Xin Zhou

Published 2026-06-02

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CC BY 4.0

Original authors: Lorenzo Sarnataro, Douglas Stryker, Zhichao Wang, Xin Zhou

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a perfectly round, solid ball of clay (a "three-ball"). Now, imagine the surface of this clay is slightly curved outward everywhere, like the skin of a drum that is pulled tight. In the world of mathematics, this is called a "strictly mean convex" shape.

The big question the authors of this paper are asking is: If you try to stretch a soap film across this clay ball, how many distinct, stable shapes can that film take?

In this context, a "soap film" is a minimal disk. It's a surface that tries to use the least amount of material possible to span a boundary. A "free boundary" means the edge of the film isn't glued down; it's allowed to slide freely along the edge of the clay ball, but it must meet the edge at a perfect 90-degree angle (like a water droplet beading up on a surface).

Here is the breakdown of their discovery, using simple analogies:

1. The Main Discovery: Finding the Hidden Shapes

The authors prove that inside any such clay ball, you can always find at least three different, non-overlapping soap films (minimal disks) that are stable.

The Analogy: Think of the clay ball as a room with a specific shape. If you throw a rubber band into the room and let it snap into the tightest possible shape, it might land in one spot. But this paper proves that no matter how you shape the room (as long as it's convex), there are actually three distinct "sweet spots" where a rubber band can snap into a perfect, stable shape without touching the others.

2. The "What If" Scenario: The Two Paths

The paper also offers a fascinating "either/or" scenario (a dichotomy) for these shapes:

Path A: The room is so simple that no closed loops (like a complete sphere floating inside) can exist. In this case, the room is guaranteed to have those three distinct soap films attached to the walls.
Path B: If the room does contain a closed loop (a floating sphere), then that sphere must be a very specific, stable type. If that happens, the math forces the existence of three distinct soap films anyway.

In short: Whether the room has floating spheres inside or not, you are guaranteed to find at least three distinct soap films attached to the boundary.

3. The Special Case: The Perfect Sphere

If the clay ball is not just convex but also has a specific type of curvature (nonnegative Ricci curvature, which includes the standard round ball in our everyday 3D space), the result is even stronger:

You are guaranteed at least three distinct, embedded free boundary minimal disks.
The authors note that for a standard round ball in our world, this means there are at least three different ways a soap film can stretch across the edge and settle into a stable, flat(ish) shape.

4. How They Did It: The "Multiplicity One" Trick

How did they prove there are three and not just one? They used a powerful mathematical tool called Min-Max Theory.

The Analogy: Imagine you are trying to find the highest mountain pass between two valleys. You don't just look at one path; you imagine all possible paths and look for the "best worst" path (the lowest high point).
The Problem: Sometimes, when you find this "best" path, the math says the solution might be "double" or "triple" (like two sheets of paper stuck together). This makes it hard to count distinct shapes.
The Breakthrough: The authors developed a new rule (a "Multiplicity One Theorem"). They proved that under their specific conditions, the math cannot produce double or triple sheets. The solution must be a single, distinct sheet. This allowed them to count the solutions one by one, proving that there are truly three separate shapes, not just one shape counted three times.

5. The Strategy: Peeling the Onion

To ensure they found a soap film attached to the wall (and not a floating sphere in the middle), they used a clever strategy:

The Flow: They imagined the boundary of the clay ball shrinking inward like a deflating balloon (Mean Curvature Flow).
The Barrier: They showed that if this shrinking process stops, it stops at a stable sphere.
The Construction: They built a mathematical "cylinder" around this stopping point to create a new, slightly different space.
The Result: By running their "search for the best path" in this new space, they proved that the solution must be a disk attached to the wall, because a floating sphere would be impossible in that specific setup.

Summary

This paper solves a long-standing puzzle in geometry. It confirms that in any convex 3D shape (like a ball), nature doesn't just offer one way to stretch a minimal surface; it offers at least three distinct, stable ways. They achieved this by inventing a new mathematical filter that ensures every solution they find is a single, unique object, allowing them to count them accurately.

Technical Summary: Existence of Free Boundary Minimal Disks in Convex Regions

Problem Statement
The paper addresses the existence and multiplicity of embedded free boundary minimal disks in compact Riemannian three-manifolds with strictly mean convex boundaries. Motivated by Yau's 1982 conjecture regarding the existence of four distinct embedded minimal spheres in $S^3$ , the authors investigate the analogous problem for three-balls ( $B^3$ ). Specifically, they seek to prove the existence of at least three distinct embedded free boundary minimal disks in any manifold diffeomorphic to $B^3$ with strictly mean convex boundary, particularly under conditions of nonnegative Ricci curvature. While the existence of at least one such disk was established by Grüter-Jost and Li, and a second by Haslhofer-Ketover, the existence of a third remains a significant open problem, even for strictly convex domains in Euclidean space.

Methodology
The authors employ a combination of geometric analysis tools, centering on a refined application of the Simon–Smith min-max theory adapted to the free boundary setting. The core methodological innovations include:

Multiplicity One Theorem: The primary analytical tool is a new multiplicity one theorem for the free boundary Simon–Smith min-max theory. The authors prove that for a homotopy class of sweepouts by disks, the resulting min-max limit consists of a collection of minimal surfaces where unstable, two-sided components appear with multiplicity one. This is achieved by solving a sequence of min-max problems for a prescribed mean curvature functional with carefully chosen functions $h$ that vanish in a neighborhood of the boundary. This allows the application of interior regularity theory while circumventing the need for developing new boundary regularity theory.
Cylindrical Manifold Construction: To ensure the min-max process yields a free boundary disk rather than a closed minimal sphere (which can occur in the interior of a convex ball), the authors utilize a construction by Song. They analyze the mean curvature flow starting from the boundary $\partial M$ . If the flow does not become extinct, it converges to stable minimal spheres. The authors then attach cylindrical ends to these stable spheres, creating a non-compact manifold. They demonstrate that the min-max width for sweepouts by disks in this cylindrical manifold remains bounded, forcing the existence of a free boundary minimal disk in the original manifold via the maximum principle.
Lusternik–Schnirelmann Theory: To establish the existence of three distinct surfaces, the authors construct three specific nontrivial sweepouts of the three-ball, parameterized by twisted bundles over real projective spaces. They utilize Lusternik–Schnirelmann theory to show that the min-max widths associated with these three sweepouts are strictly increasing ( $L(P_1) < L(P_2) < L(P_3)$ ), provided the manifold contains only finitely many minimal surfaces.

Key Results
The paper establishes the following main theorems:

Theorem 1.1 (Multiplicity of Solutions): Let $(M^3, \partial M, g)$ $(M^{3}, \partial M, g)$ be a Riemannian three-ball with strictly mean convex boundary, assuming no degenerate stable minimal spheres or free boundary minimal disks exist. Then, there exist at least three distinct embedded submanifolds which are either minimal spheres or free boundary minimal disks.
- Corollary: If $(M^3, g)$ is a strictly convex domain with nonnegative Ricci curvature (e.g., a compact convex ball in $\mathbb{R}^3$ ), it admits at least three distinct embedded free boundary minimal disks.
Theorem 1.2 (Structural Dichotomy): For any Riemannian three-ball with strictly mean convex boundary, there exists at least one embedded free boundary minimal disk. Furthermore, the manifold satisfies one of two alternatives:
1. It contains no closed immersed minimal surfaces; or
2. It contains an embedded stable minimal sphere.
Theorem 1.3 (Multiplicity One Theorem): In a compact Riemannian three-manifold with smooth boundary, for a homotopy class of sweepouts with width strictly greater than the boundary area, the min-max limit consists of pairwise disjoint minimal surfaces. If a component is two-sided and unstable, its multiplicity is exactly one. If a component is one-sided, its connected double cover is stable.

Significance and Claims
The authors claim that this work resolves the analogue of Yau's conjecture for three-balls under specific non-degeneracy assumptions. The significance of the paper lies in:

Resolving the Multiplicity Problem: It proves the existence of at least three distinct free boundary minimal disks in strictly convex domains with nonnegative Ricci curvature, a result that was previously open.
Technical Advancement: The paper provides a robust multiplicity one theorem for the free boundary Simon–Smith setting. A key technical contribution is the observation that the prescribed mean curvature functions can be chosen to have support strictly in the interior (away from the boundary), thereby allowing the direct application of existing interior regularity theories without the arduous task of developing new boundary regularity estimates.
Structural Insight: The dichotomy result (Theorem 1.2) clarifies the relationship between the existence of free boundary disks and the presence of closed minimal spheres or stable minimal spheres within the manifold.

The paper emphasizes that while the result holds for "bumpy" metrics (generic metrics where all minimal surfaces are non-degenerate) and strictly convex domains with nonnegative Ricci curvature, the general case for strictly convex domains in $\mathbb{R}^3$ without the non-degeneracy assumption remains an open problem. The results rely on the assumption that the manifold does not contain degenerate stable minimal spheres or free boundary minimal disks.