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Existence of two embedded minimal spheres in $S^3$ with an arbitrary metric

The paper proves that any Riemannian metric on the 3-sphere $S^3$ admits at least two embedded minimal spheres, utilizing an iterative scheme of relative min-max constructions.

Original authors: Zhichao Wang, Xin Zhou

Published 2026-07-10

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: 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 the universe's shape, specifically a perfect 3-dimensional sphere (like a balloon, but with an extra dimension we can't quite see). Now, imagine stretching and squishing this balloon with a weird, bumpy texture. This texture is what mathematicians call a "Riemannian metric." It's like the balloon is made of rubber that has been pulled tighter in some spots and looser in others.

For decades, mathematicians have been hunting for "minimal spheres" on these bumpy balloons. Think of a minimal sphere as a soap bubble that has settled into its most relaxed, lowest-energy state. It's the perfect, smoothest loop you can draw on that bumpy surface that doesn't want to shrink or expand any further.

The Big Discovery
In 1982, a famous mathematician named S. T. Yau asked a tough question: "If you take any bumpy 3-sphere, can you always find four distinct, perfect soap-bubble spheres hidden inside it?"

For over 40 years, the best anyone could do was prove that there is at least one such sphere. It was like finding a single hidden treasure on a map, but the map promised four.

Now, two researchers, Zhichao Wang and Xin Zhou, have stepped up and proven a major step forward. They have shown that on any bumpy 3-sphere, no matter how weird the texture, there are guaranteed to be at least two distinct, embedded minimal spheres. They haven't found all four yet, but they've proven that the "one is enough" idea is wrong. There is always a second one hiding there.

Why Was This So Hard?
You might wonder, "If there's one, why not just find another?" The problem is a tricky mathematical glitch called "multiplicity."

Imagine you are looking for a second soap bubble. You use a special mathematical tool (called a "min-max" construction) to hunt for it. But sometimes, this tool gets confused. Instead of finding a new bubble, it just finds the same bubble you already found, but it counts it twice because it's "stacked" on top of itself. It's like looking in a mirror and thinking you see a second person, but it's just a reflection.

Previous attempts to find a second sphere relied on the balloon having a very specific, nice texture (positive Ricci curvature). But the real world (and the math problem) allows for any texture, even the messy, weird ones. The authors had to prove that even in the messiest scenarios, the "double-counting" glitch doesn't happen, and a truly second, distinct sphere exists.

How They Solved It: The "Neck" Trick
To prove this, the authors used a clever strategy involving a "contradiction." They said, "Let's pretend there is only one minimal sphere on the whole balloon. We'll try to break that idea."

The Setup: They imagined a family of surfaces that sweep across the balloon. Think of these as a series of slices, like cutting a loaf of bread, but the slices can wiggle and change shape.
The "Neck" Construction: They built a complex family of these slices using a parameter space (a multi-dimensional triangle). In the middle of this triangle, they created surfaces that look like two bubbles connected by a thin "neck" (like a dumbbell).
The Area Trap: They calculated the total area of these shapes. If there were only one minimal sphere, the math says the area of these shapes should be exactly a certain amount. But, by making the "neck" between the two parts very thin and short, they showed that the total area could be made strictly less than what the "only one sphere" theory predicted.
The Contradiction: Because they found a shape with less area than the theory allowed, the theory that "there is only one sphere" must be false. Therefore, a second sphere must exist to "absorb" that extra area.

What They Didn't Do
It's important to know what this paper doesn't say. They did not prove there are four spheres (Yau's original guess). They only proved there are at least two. They also didn't assume the balloon was perfectly round or had a nice, smooth texture; their proof works for arbitrary (completely random and messy) textures.

The Bottom Line
This isn't a simulation or a guess; it is a rigorous mathematical proof. Wang and Zhou have successfully navigated the "multiplicity" trap that stumped mathematicians for decades. They have shown that no matter how you stretch or bumpy up a 3-sphere, you can never hide the fact that there are at least two distinct, perfect minimal spheres waiting to be found. It's a massive leap forward in understanding the hidden geometry of our universe.

Technical Summary: Existence of Two Embedded Minimal Spheres in $S^3$ with an Arbitrary Metric

Problem Statement
The paper addresses a long-standing problem posed by S. T. Yau in 1982 (Problem 89 in [44]), which asks for the proof that any Riemannian manifold diffeomorphic to the 3-sphere ( $S^3$ ) contains at least four distinct embedded minimal spheres. While Simon and Smith [36] established the existence of at least one such sphere over 40 years ago using min-max theory, the existence of a second distinct embedded minimal sphere for arbitrary Riemannian metrics remained unproven. Previous results establishing multiple spheres (e.g., White [43], Haslhofer–Ketover [17]) relied on restrictive assumptions, such as positive Ricci curvature or "bumpy" metrics. This paper aims to remove these curvature assumptions and prove the existence of at least two distinct embedded minimal spheres for any Riemannian metric on $S^3$ .

Methodology
The authors employ a proof by contradiction combined with an iterative scheme of relative min-max constructions. The core methodology involves the following steps:

Assumption of Uniqueness: The proof assumes that $(S^3, g)$ contains exactly one embedded minimal sphere, denoted $\Sigma_0$ . Based on prior work [40], if $\Sigma_0$ is unique, it must be weakly stable and possess an "expanding neighborhood" (foliated by constant mean curvature surfaces with mean curvature vectors pointing away from $\Sigma_0$ ).
Construction of Multi-Parameter Sweepouts: The authors construct a continuous family of piecewise-smooth embedded spheres, $\Phi_\Lambda: \Delta_k \to \mathcal{E}$ , parameterized by a $k$ -dimensional simplex $\Delta_k$ .
- Foliation: They utilize a mean convex foliation $\Upsilon$ of $S^3$ centered at $\Sigma_0$ .
- Neck Insertion: For a parameter vector $x = (x_1, \dots, x_k) \in \Delta_k$ , the map $\Phi_\Lambda$ creates a sphere by taking the union of slices from the foliation $\Upsilon(x_i)$ and connecting them with "necks."
- Radius Control: The radii of these necks, denoted $\xi_i$ , are defined recursively based on the distance between adjacent parameters $x_i$ and $x_{i+1}$ and a large parameter $\Lambda$ . Specifically, $\xi_i = e^{-\Lambda(x_{i+1}-x_i)}(1-\xi_{i-1})$ . This construction ensures that if two parameters are close, the connecting neck has a large radius but small height, resulting in a uniformly small area contribution.
Area Estimates and Smoothing:
- The authors derive precise area bounds for the constructed family. Crucially, they show that for sufficiently large $k$ and appropriate $\Lambda$ , the area of the family $\Phi_\Lambda(x)$ is strictly less than $k H^2(\Sigma_0)$ in the interior of the simplex, while on the boundary, the area is strictly less than $(k-1)H^2(\Sigma_0)$ .
- They address the issue of piecewise-smoothness by smoothing the family to obtain a map into the space of smooth embedded spheres, preserving the strict area upper bounds.
Iterative Relative Min-Max:
- Using the Relative Simon–Smith min-max theory (Theorem 2.2), the authors define min-max values $L_k$ associated with sweepouts over sub-simplices.
- They employ a Lusternik–Schnirelmann type argument. Assuming uniqueness, they inductively prove that the min-max value for the $k$ -th sweepout must equal $k H^2(\Sigma_0)$ .
- Topological Obstruction: A key component is Lemma 6.1, a topological lemma ensuring that if the min-max value were strictly less than the boundary values (which would imply the existence of a new minimal sphere), the domain could be retracted in a way that contradicts the homological properties of the sweepout.
Contradiction:
- The construction ensures that for large $k$ , the area of the sweepout in the interior is strictly less than $k H^2(\Sigma_0)$ .
- However, the inductive argument under the uniqueness assumption forces the min-max value to be exactly $k H^2(\Sigma_0)$ .
- This contradiction implies that the assumption of uniqueness is false.

Key Contributions

Removal of Curvature Assumptions: The paper extends the existence of multiple minimal spheres to arbitrary Riemannian metrics on $S^3$ , a significant generalization over previous results that required positive Ricci curvature or bumpy metrics.
Iterative Relative Min-Max Scheme: The authors develop a novel iterative scheme using relative min-max constructions on multi-parameter families of spheres. This approach overcomes the "integer multiplicity" obstacle, where the Simon–Smith theory might otherwise produce a single minimal sphere with higher multiplicity (e.g., $\Sigma_0$ counted twice).
Neck Construction and Area Control: The specific construction of the multi-parameter family with controlled neck radii allows for strict area upper bounds that are essential for the contradiction argument. This handles the difficulty of extending the map continuously to the entire simplex while maintaining area constraints.

Results
The main result is stated as Theorem A (restated as Theorem 6.3):

There exist at least two distinct embedded minimal spheres in $(S^3, g)$ for any arbitrary Riemannian metric $g$ .

This result confirms the first step toward Yau's conjecture of four spheres, establishing the existence of a second distinct sphere without geometric restrictions on the metric.

Significance
The paper claims this work marks the first time more than one embedded minimal sphere has been constructed for general metrics on $S^3$ . It advances the general problem significantly from the initial result of Simon and Smith [36] (which found one sphere) by breaking the barrier of integer multiplicity for arbitrary metrics. The authors note that while their previous work [40] resolved the conjecture for metrics with positive Ricci curvature or bumpy metrics using a multiplicity-one theorem, this paper extends the reach to the general case. The methods developed, particularly the iterative relative min-max construction and the handling of area bounds without Ricci curvature assumptions, are presented as tools that could potentially be applied to investigate similar problems in geometric topology without requiring specific curvature conditions.

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