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An almost-almost-Schur lemma on the 3-sphere

Tobias König, Jonas W. Peteranderl

TL;DR

This work extends a quantitative stability result for reverse curvature inequalities to the 3-sphere, recasting the problem in terms of conformal factors and the conformally invariant functionals $F_1$ and $F_2$. It proves a sharp deficit bound for the Andrews–De Lellis–Topping inequality on $\mathbb S^3$, with proximity measured by a two-term distance after optimizing over scalings and Möbius transformations. The authors also establish a reverse $\sigma_2$-curvature stability and derive an interpolation family of reverse inequalities, all proved via a two-step Bianchi–Egnell strategy that combines global-to-local reduction with a detailed local bound grounded in Hessian analysis and frequency decomposition. The results connect to almost-Schur type questions and yield stability statements for related interpolation inequalities, enriching the landscape of stability in conformal geometry and reverse Sobolev-type inequalities.

Abstract

In a recent preprint, Frank and the second author proved that if a metric on the sphere of dimension $d>4$ almost minimizes the total $σ_2$-curvature in the conformal class of the standard metric, then it is almost the standard metric (up to Möbius transformations). This is achieved quantitatively in terms of a two-term distance to the set of minimizing conformal factors. We extend this result to the case $d=3$. While the standard metric still minimizes the total scalar curvature for $d=3$, it maximizes the total $σ_2$-curvature, which turns the related functional inequality into a reverse Sobolev-type inequality. As a corollary of our result, we obtain quantitative versions for a family of interpolation inequalities including the Andrews--De Lellis--Topping inequality on the $3$-sphere. The latter is itself a stability result for the well-known Schur lemma and is therefore called almost-Schur lemma. This makes our stability result an almost-almost-Schur lemma.

An almost-almost-Schur lemma on the 3-sphere

TL;DR

This work extends a quantitative stability result for reverse curvature inequalities to the 3-sphere, recasting the problem in terms of conformal factors and the conformally invariant functionals and . It proves a sharp deficit bound for the Andrews–De Lellis–Topping inequality on , with proximity measured by a two-term distance after optimizing over scalings and Möbius transformations. The authors also establish a reverse -curvature stability and derive an interpolation family of reverse inequalities, all proved via a two-step Bianchi–Egnell strategy that combines global-to-local reduction with a detailed local bound grounded in Hessian analysis and frequency decomposition. The results connect to almost-Schur type questions and yield stability statements for related interpolation inequalities, enriching the landscape of stability in conformal geometry and reverse Sobolev-type inequalities.

Abstract

In a recent preprint, Frank and the second author proved that if a metric on the sphere of dimension almost minimizes the total -curvature in the conformal class of the standard metric, then it is almost the standard metric (up to Möbius transformations). This is achieved quantitatively in terms of a two-term distance to the set of minimizing conformal factors. We extend this result to the case . While the standard metric still minimizes the total scalar curvature for , it maximizes the total -curvature, which turns the related functional inequality into a reverse Sobolev-type inequality. As a corollary of our result, we obtain quantitative versions for a family of interpolation inequalities including the Andrews--De Lellis--Topping inequality on the -sphere. The latter is itself a stability result for the well-known Schur lemma and is therefore called almost-Schur lemma. This makes our stability result an almost-almost-Schur lemma.

Paper Structure

This paper contains 14 sections, 9 theorems, 91 equations.

Table of Contents

  1. Introduction and main results
  2. An almost-Schur lemma
  3. An extension by Ge and Wang
  4. Functional formulation on $\mathbb S^3$
  5. An almost-almost-Schur lemma
  6. The reverse $\sigma_2$-curvature inequality
  7. An interpolation family of reverse inequalities and their stability
  8. Some context and related works
  9. Proof strategy
  10. The Bianchi--Egnell strategy
  11. Proof of the main results
  12. Global-to-local reduction
  13. Orthogonality conditions and comparability between $W^{1,2}$ and $W^{1,4}$
  14. Local bound

Key Result

Theorem 1.1

There is a constant $c_{ADT}>0$ such that for all $u\in C^\infty(\mathbb S^3)$ with $u>0$ and $\sigma_1(u)>0$ we have where the infimum is taken over all $\lambda\in \mathbb{R}$ and Möbius transformations $\Psi:\mathbb S^3\to\mathbb S^3$.

Theorems & Definitions (17)

  • Theorem 1.1: Quantitative stability for \ref{['eq:ADT_intro_functional']}
  • Theorem 1.2: Quantitative stability for \ref{['eq:reverse_sigma2']}
  • Corollary 1.4: Interpolated stability
  • Proposition 2.1: Global-to-local reduction
  • Proposition 2.2: Local bound
  • proof : Proof of Theorem \ref{['thm:revsigma']}
  • proof : Proof of Corollary \ref{['cor:interpolation_stability']}
  • proof : Proof of Proposition \ref{['prop:glob2locrev']}
  • Lemma 3.1
  • proof
  • ...and 7 more