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Existence of Yamabe stability optimizers

This paper establishes the existence of Yamabe stability optimizers on closed Riemannian manifolds of dimension at least three by proving that the stability constant remains strictly below the one-bubble threshold under specific dimensional, geometric, or positive-mass-type conditions.

Original authors: João Henrique Andrade, Tobias König, Jesse Ratzkin, Juncheng Wei

Published 2026-08-12

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CC BY 4.0

Original authors: João Henrique Andrade, Tobias König, Jesse Ratzkin, Juncheng Wei

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 not as a void of empty space, but as a giant, stretchy rubber sheet. In the world of geometry, this sheet is called a "manifold," and it can be shaped like a sphere, a donut, or something far more twisted and complex. Mathematicians have long been obsessed with a specific question: Can we stretch or squash this sheet just right so that its "curvature" (how much it bends) is the same everywhere? This is the famous Yamabe problem. Think of it like trying to smooth out a crumpled piece of paper until it's perfectly flat, or inflating a balloon until every inch feels exactly the same.

To solve this, mathematicians use a special "scorecard" called the Yamabe inequality. This scorecard tells you how close a shape is to being perfectly smooth. If the score is perfect, the shape is a "Yamabe optimizer"—the gold standard of smoothness. But what happens if you're almost perfect, but not quite? How far off are you? This is the question of "stability." It's like asking: if you're just a tiny bit out of tune on a guitar, how much does the sound wobble? For a long time, we knew the scorecard existed, but we didn't know if there was a specific "worst-case scenario" function that pushed the scorecard to its absolute limit. We knew the limit existed, but we didn't know if anyone could actually reach it.

This paper, written by Andrade, König, Ratzkin, and Wei, dives deep into that "almost perfect" zone. They prove that for most shapes (specifically, closed, smooth surfaces in three or more dimensions that aren't perfect spheres), there is a specific function that hits the stability limit. They call these functions "stability optimizers." It's like finding the exact, single note that makes a guitar string wobble the most before it snaps.

The authors discovered that finding this "worst-case note" depends on two main rules. First, the shape can't be a perfect sphere (which is a special case already solved by others). Second, the shape has to pass a specific "mass test." Imagine the shape has a hidden weight or "mass" hidden inside its geometry. The paper shows that if this hidden mass is heavy enough compared to the shape's natural "wobble," then the stability optimizer exists. If the mass is too light, the optimizer might vanish into thin air.

The team used a clever strategy involving "bubbles." Imagine trying to smooth out a crumpled sheet by blowing bubbles into it. Sometimes, the sheet tries to fix itself by forming a giant bubble that pops off. The authors proved that for most shapes, the sheet can't just rely on popping a bubble to bypass the system. They showed that in high dimensions (six and up) or for shapes that aren't perfectly flat in a local sense, the "bubble" strategy fails to beat the stability limit. In lower dimensions (three to five) or for very flat shapes, they found a new rule: the hidden mass must be strong enough to stop the bubble from winning.

In short, the paper proves that for a vast range of geometric shapes, there is a definitive, mathematical "champion" of instability. It's a rigorous proof that the limit isn't just a theoretical ghost; it's a real, reachable point in the landscape of shapes. The authors didn't just guess; they built a mathematical bridge using test functions (imaginary shapes) and showed that the bridge holds firm, proving that these optimizers definitely exist under the right conditions.

Technical Summary: Existence of Yamabe Stability Optimizers

Problem Statement
The paper investigates the quantitative stability of the Yamabe inequality on closed Riemannian manifolds $(M, g)$ of dimension $n \geqslant 3$ with positive Yamabe invariant $Y(M, [g]) > 0$ . The Yamabe problem establishes that every conformal class contains a metric of constant scalar curvature, minimizing the Yamabe quotient $Q(u)$ . The authors address the stability of this inequality, specifically asking whether the "best stability constant" in the Bianchi-Egnell-type inequality is attained.

The stability inequality, established by Engelstein, Neumayer, and Spolaor, states that for a stability exponent $\gamma \geqslant 2$ , there exists a constant $c_\gamma(M, g) > 0$ such that:
$E[u](\gamma-2)/2 \left( E[u] - Y(M, [g])\|u\|_{L^p(M)}^2 \right) \geqslant c_\gamma(M, g) \, \text{dist}(u, \mathcal{O})^\gamma$
where $E[u]$ is the Yamabe energy, $\mathcal{O}$ is the set of Yamabe optimizers, and $\text{dist}$ is the distance in the energy norm. The central question is whether the infimum defining $c_\gamma(M, g)$ is achieved by a function $u \in H^1(M) \setminus \mathcal{O}$ (an "optimizer").

Methodology
The authors adapt the concentration-compactness strategy previously used for the Euclidean Sobolev inequality (specifically the work of the second author on the sphere) to the general manifold setting. The proof relies on analyzing minimizing sequences for the stability quotient $S_\gamma(u)$ .

Threshold Analysis: The attainability of the constant depends on comparing $c_\gamma(M, g)$ against two critical thresholds:
- Local Threshold ( $c_{\text{loc}}^\gamma$ ): The limit of $S_\gamma$ along sequences converging to an optimizer.
- Bubble Threshold ( $c_{\text{bub}}^\gamma$ ): The limit of $S_\gamma$ along sequences that concentrate and form bubbles (weakly converging to zero).
  The authors prove that if $c_\gamma(M, g)$ is strictly below both thresholds, minimizing sequences converge strongly to an optimizer.
Compactness of Optimizers: A crucial technical step involves proving the compactness of the set of $L^p$ -normalized Yamabe optimizers $\mathcal{O}_1$ when $Y(M, [g]) < Y(S^n, [g_\circ])$ . This contrasts with the round sphere case where the set of optimizers is non-compact.
Asymptotic Analysis: To verify the bubble threshold condition ( $c_\gamma < c_{\text{bub}}^\gamma$ ), the authors construct specific test functions and perform asymptotic expansions of the stability quotient. They distinguish between two regimes based on dimension and conformal geometry:
- Non-Locally Conformally Flat (LCF) manifolds ( $n \geqslant 6$ ): They utilize truncated Aubin-Talenti bubbles. The expansion reveals that the energy correction (driven by the Weyl curvature) dominates the stability mass term, ensuring the strict inequality.
- Locally Conformally Flat manifolds or low dimensions ( $3 \leqslant n \leqslant 5$ ): The Weyl curvature vanishes or the energy correction is of lower order. Here, the authors employ Schoen's Green function modified test functions. They derive a "mass dominance condition" comparing the Green function mass ( $m_{x_0}$ ) against a "stability mass" coefficient ( $m_\infty(x_0)$ ).

Key Contributions and Results

Existence Theorem (Theorem 1.2): The paper establishes that a stability optimizer exists if the best constant $c_\gamma(M, g)$ is strictly less than both the local threshold $c_{\text{loc}}^\gamma(M, g)$ and the bubble threshold $c_{\text{bub}}^\gamma(M, g)$ . This provides a unified framework for existence that applies to general manifolds, distinct from the specific two-bubble threshold found in the round sphere case.
Bubble Threshold Verification (Theorem 1.3): For manifolds with $n \geqslant 6$ that are not locally conformally flat, the authors prove that $c_\gamma(M, g) < c_{\text{bub}}^\gamma(M, g)$ holds unconditionally. This is achieved by showing that the negative energy correction from the Weyl tensor (of order $\mu^4$ ) dominates the positive stability mass term (of order $\mu^{n-2}$ ).
Mass Dominance Condition (Theorem 1.4): In the complementary cases (locally conformally flat manifolds or dimensions $3 \leqslant n \leqslant 5$ ), the authors introduce a new sufficient condition for the strict inequality. They prove that if a "mass dominance condition" holds at some point $x_0$ (where the Green function mass sufficiently outweighs the stability mass coefficient), then $c_\gamma(M, g) < c_{\text{bub}}^\gamma(M, g)$ .
Corollary 1.5: Combining the above, the paper provides explicit geometric criteria (involving dimension, local conformal flatness, and the mass dominance condition) under which stability optimizers are guaranteed to exist.

Significance
The paper resolves the open question of whether stability optimizers exist for the Yamabe inequality on general closed manifolds with positive scalar curvature. It highlights a fundamental difference between the spherical case and the non-spherical case: while the sphere requires excluding a "two-bubble" concentration scenario, general manifolds only require excluding a "one-bubble" scenario due to the compactness of the optimizer set.

The work introduces the concept of a "stability mass" functional, which plays a role analogous to the Green function mass in the classical Yamabe problem but acts as a competing term in the stability quotient. The authors demonstrate that in low dimensions or locally conformally flat settings, the positivity of the Green function mass alone is insufficient; a quantitative comparison (the mass dominance condition) is necessary to ensure stability. This refines the understanding of quantitative stability in geometric analysis, moving beyond the classical Yamabe problem to the finer structure of the stability inequality.

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