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Sharp stability of Alexandrov's theorem for $C^1$ domains in the small-excess regime

This paper establishes a sharp quantitative stability result for Alexandrov's theorem in arbitrary dimensions, proving that for bounded $C^1$ domains with small excess, both the geometric deviation from a ball and the excess are controlled by the optimal $L^2$ -oscillation of the mean curvature.

Original authors: Alessio Figalli, Yi Ru-Ya Zhang

Published 2026-06-12

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Alessio Figalli, Yi Ru-Ya Zhang

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

The Big Picture: The "Soap Bubble" Rule

Imagine you have a soap bubble. Physics tells us that if a soap bubble has a perfectly constant surface tension (which creates a constant "mean curvature"), it must be a perfect sphere. This is a famous mathematical rule called Alexandrov's Theorem.

But what if the bubble isn't perfect? What if the surface tension wobbles just a tiny bit? Is the bubble still close to being a sphere, or could it be a weird, lumpy shape that just happens to have similar tension?

This paper answers that question. It proves that if a shape is "almost" a sphere (in a specific mathematical sense called "small excess") and its surface tension is "almost" constant, then the shape must be very, very close to a perfect sphere. Furthermore, the authors calculate exactly how close it is, providing a precise mathematical ruler to measure the difference.

The Main Characters

The Shape ( $E$ ): Think of this as a blob of clay or a soap bubble. It has a boundary (skin).
The Perfect Sphere ( $B$ ): The gold standard. The perfect ball.
The "Excess" ($Exc(E)$): Imagine you have a perfect sphere. Now, imagine your blob is slightly squashed or stretched. The "excess" is a number that measures how much extra surface area your blob has compared to a perfect sphere of the same volume. If the excess is zero, it's a perfect sphere. If it's small, it's almost a sphere.
The "Mean Curvature" ( $H$ ): This is the mathematical way of describing how "curved" the surface is at every point. For a perfect sphere, this number is the same everywhere.
The "Oscillation": This measures how much the curvature wobbles. If the curvature is 5 everywhere, the oscillation is 0. If it wiggles between 4.9 and 5.1, the oscillation is small.

The Problem: Why is this hard?

In the past, mathematicians could prove this "stability" only if the shape was already very close to a sphere in a very strict way (like a graph drawn over a sphere). But real-world shapes (or shapes in higher dimensions) can be weird. They might have "tentacles" sticking out, or they might be shaped like a starfish.

The authors wanted to prove this rule works for any shape that is reasonably smooth (called a $C^1$ domain), even if it looks a bit messy, as long as it doesn't have too much extra surface area (small excess).

The Solution: How they did it

The authors used a clever four-step strategy to tame the messy shapes:

1. The "Tentacle" Trimming
Imagine your blob has long, thin fingers or "tentacles" sticking out far away from the center. These are annoying to analyze.

The Trick: The authors proved that if the "excess" is small, these tentacles can't be very long or numerous. They essentially "cut off" the tentacles, leaving a core shape that sits neatly in a thin ring (an annulus) around the perfect sphere. They showed that the amount of surface area they cut off is tiny and controlled by the wobble in the curvature.

2. The "Star-Shaped" Makeover
Once the tentacles are gone, the remaining shape is still a bit lumpy.

The Trick: They used a mathematical technique called "star-shaped rearrangement." Imagine taking the blob and, for every direction you look, stretching or shrinking the distance from the center so that the volume stays the same, but the shape becomes "star-like" (no holes or weird indentations). This turns the messy blob into a shape that looks like a graph over a sphere, which is much easier to study mathematically.

3. The "Spectral Gap" (The Bouncing Ball)
Now they had a shape that looked like a slightly bumpy sphere. They needed to prove that if the curvature is almost constant, the bumps must be small.

The Analogy: Think of the surface of the sphere as a trampoline. If you push down on it (change the curvature), the trampoline wants to bounce back to being flat. The "spectral gap" is a mathematical way of saying the trampoline is very stiff; it doesn't allow for big, floppy waves unless you push really hard. The authors adapted a famous argument (by Fuglede) to show that for these specific shapes, the "stiffness" is high enough to force the shape to stay close to a sphere.

4. The "Polyhedral" Approximation
To make the math rigorous, they couldn't just assume the shape was smooth. They approximated the smooth shape with a shape made of flat faces (like a geodesic dome or a soccer ball).

The Trick: They proved that if the rule works for these blocky, flat-faced shapes, it also works for the smooth shapes, provided the blocks are small enough. This allowed them to use powerful tools from geometry that usually only work on flat surfaces.

The Result: The "Sharp" Ruler

The paper concludes with a precise inequality (a mathematical formula). It says:

The "Distance" to a Sphere $\le$ Constant $\times$ (The Wobble in Curvature) $^2$

Distance: This includes two things: how much extra surface area the shape has (Excess) and how much the shape differs from a perfect ball (Symmetric Difference).
Wobble: This is the $L^2$ -oscillation of the mean curvature (how much the curvature varies).
Squared: The most important part is the square. This means if you halve the wobble in the curvature, the shape gets four times closer to being a perfect sphere. This is called "sharp" stability because it's the best possible rate; you can't get a better relationship than this.

Why "C1" Matters

The paper specifically deals with $C^1$ domains. In everyday language, this means the shape is smooth enough that it has a well-defined tangent plane everywhere (no sharp corners like a cube, but it doesn't need to be perfectly smooth like a polished marble). The authors show that even with this relatively modest requirement of smoothness, the "soap bubble" rule holds true with this perfect, sharp precision.

Summary

If you have a shape that is:

Roughly the size of a ball.
Has very little extra surface area (it's not too wrinkly).
Has a surface tension that is almost the same everywhere.

Then, this shape must be extremely close to a perfect sphere. The authors didn't just say "it's close"; they gave a precise formula showing that the closeness improves quadratically as the surface tension becomes more uniform. They achieved this by cutting off the weird "tentacles," smoothing the shape into a star, and using a rigid "trampoline" argument to prove the bumps can't be big.

Technical Summary: Sharp Stability of Alexandrov's Theorem for $C^1$ Domains in the Small-Excess Regime

Problem Statement
Alexandrov's Soap Bubble Theorem asserts that the only compact, connected, embedded hypersurfaces in $\mathbb{R}^n$ with constant mean curvature are spheres. A fundamental question in geometric analysis concerns the stability of this result: if a hypersurface has "nearly" constant mean curvature, is it quantitatively close to a sphere?

Previous stability results have largely relied on parametric assumptions, requiring the set to be a small perturbation of a sphere (e.g., a radial graph over $S^{n-1}$ ) or possessing high regularity ( $C^2$ ). Furthermore, many existing estimates depend on constants that are not purely dimensional or require strong perimeter bounds to prevent "bubbling" (the formation of disjoint spheres). This paper addresses the stability of Alexandrov's theorem in a genuinely non-parametric regime for bounded $C^1$ open sets in arbitrary dimensions, specifically within a small-excess regime. The goal is to establish a sharp quantitative estimate where the deviation from a sphere is controlled solely by the $L^2$ -oscillation of the mean curvature, with constants depending only on the dimension $n$ .

Methodology
The proof combines several advanced techniques from geometric measure theory and spectral analysis:

BV Spectral Gap Argument: The authors adapt Fuglede's classical spectral-gap argument for the isoperimetric inequality to sets of finite perimeter. This involves a delicate analysis of functions of bounded variation (BV) on the sphere, utilizing a spectral gap for the Laplacian to control the $L^2$ -norm of perturbations by their gradient energy.
Star-Shaped Rearrangement: To handle non-parametric sets, the authors employ a star-shaped rearrangement. This transforms a general set of finite perimeter into a star-shaped set (a radial graph) while preserving volume and the weighted barycenter. Crucially, they prove that this rearrangement does not significantly increase the perimeter or the curvature energy in the small-excess regime.
Tentacle Control and Truncation: The proof isolates "tentacles" (regions of the boundary far from the unit sphere) and shows that their contribution to the perimeter and curvature is negligible when the excess is small. This is achieved using coarea formulas and Michael-Simon type inequalities to bound the area of these regions by the $L^2$ -oscillation of the curvature.
Polyhedral Approximation: A key technical step involves approximating the $C^1$ domain by polyhedral sets. This allows the authors to handle the non-graphical regions of the boundary and manage the singular parts of the curvature measure (arising from the cutting interfaces) rigorously.
Leveling Argument: For the non-graphical part of the boundary, a "leveling" argument is used to reduce the problem to an annular configuration where the spectral gap estimates apply.

Key Contributions and Results
The main result (Theorem 1.3) establishes a sharp quantitative stability estimate. Let $E \subset \mathbb{R}^n$ be a bounded $C^1$ open set with volume equal to the unit ball $|B|$ , small excess $\text{Exc}(E)$ , and scalar distributional mean curvature $H_{\partial E} \in L^2(\partial E)$ . Then, up to a translation, the following inequality holds for any $\mu \in \mathbb{R}$ :

$\text{Exc}(E) + |E \Delta B|^2 + |\mu - (n-1)|^2 \leq C(n) \|H_{\partial E} - \mu\|_{L^2(\partial E)}^2$

where:

$\text{Exc}(E)$ is the excess, measuring the deviation of the normal vector from the radial direction.
$|E \Delta B|$ is the symmetric difference (volume distance) from the unit ball.
$C(n)$ is a constant depending only on the dimension $n$ .

Significance and Claims
The paper claims several significant advancements over prior literature:

Non-Parametric Setting: Unlike previous works (e.g., [27, 32, 17]) that required the set to be a radial graph over the sphere, this result applies to arbitrary bounded $C^1$ open sets. The $C^1$ regularity is used only as a qualitative technical ingredient to ensure the existence of polyhedral approximations and to handle the projection of tangency sets; the final constants do not depend on the $C^1$ norm of the boundary.
Sharp Scaling: The estimate retains the sharp quadratic scaling ( $L^2$ -oscillation squared) found in the parametric case, which is optimal.
Dimensional Constants: The stability constant $C(n)$ depends solely on the dimension, avoiding dependence on higher-order norms of the boundary or specific geometric constraints like uniform interior ball conditions found in earlier moving plane method results.
Generalization of Fuglede's Argument: The work successfully extends Fuglede's spectral-gap technique to the setting of sets of finite perimeter and star-shaped rearrangements, bridging the gap between smooth geometric analysis and the theory of sets of finite perimeter.

The authors note that while the $C^1$ assumption is necessary for the proof's technical steps (specifically for the polyhedral approximation and the handling of the radial tangency set), the result represents a sharp stability estimate in a regime where the boundary is not assumed a priori to be a graph. The result also connects with compactness theories (e.g., [15]) to exclude bubbling phenomena under appropriate perimeter and diameter bounds, ensuring convergence to a single unit ball.