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Linear Quantitative Rigidity for Almost-CMC Surfaces

Yuchen Bi, Jie Zhou

TL;DR

The paper proves a linear quantitative rigidity result for almost-CMC spheres in $\mathbb{R}^3$: under a sub-two-sphere Willmore bound and a small $L^2$ CMC defect, an almost-CMC surface is close to the round sphere with linear control in the $W^{2,2}$-distance of the conformal parametrization and the $L^\infty$-norm of the conformal factor. The approach linearizes about the round sphere, derives a precise quadratic expansion of the governing energy, and uses spectral properties of the sphere together with elliptic regularity to bootstrap a sharp $W^{2,2}$-estimate controlled by the defect. The results extend to integral $2$-varifolds of unit density and admit an area-based reformulation via a two-sphere-threshold Willmore bound, leveraging a quantitative Alexandrov theorem. Overall, the work provides a robust, perturbative stability framework for near-spherical CMC configurations with potential applications to geometric measure theory and variational problems involving Willmore-type energies.

Abstract

We prove a quantitative rigidity result for almost constant mean curvature spheres in $\mathbb{R}^3$. Under a sub--two--sphere Willmore bound and a small $L^2$--CMC defect, we show that an almost--CMC surface is close to the round sphere, with linear control of the $W^{2,2}$--distance of the parametrization and the $L^\infty$--norm of the conformal factor. An analogous statement holds under an a priori area bound below that of two spheres.The proof relies on a linearized analysis around the sphere. A previously established qualitative rigidity result provides the initial closeness required to enter the perturbative regime. The estimate further extends to integral $2$--varifolds of unit density using known regularity and density results.

Linear Quantitative Rigidity for Almost-CMC Surfaces

TL;DR

The paper proves a linear quantitative rigidity result for almost-CMC spheres in : under a sub-two-sphere Willmore bound and a small CMC defect, an almost-CMC surface is close to the round sphere with linear control in the -distance of the conformal parametrization and the -norm of the conformal factor. The approach linearizes about the round sphere, derives a precise quadratic expansion of the governing energy, and uses spectral properties of the sphere together with elliptic regularity to bootstrap a sharp -estimate controlled by the defect. The results extend to integral -varifolds of unit density and admit an area-based reformulation via a two-sphere-threshold Willmore bound, leveraging a quantitative Alexandrov theorem. Overall, the work provides a robust, perturbative stability framework for near-spherical CMC configurations with potential applications to geometric measure theory and variational problems involving Willmore-type energies.

Abstract

We prove a quantitative rigidity result for almost constant mean curvature spheres in . Under a sub--two--sphere Willmore bound and a small --CMC defect, we show that an almost--CMC surface is close to the round sphere, with linear control of the --distance of the parametrization and the --norm of the conformal factor. An analogous statement holds under an a priori area bound below that of two spheres.The proof relies on a linearized analysis around the sphere. A previously established qualitative rigidity result provides the initial closeness required to enter the perturbative regime. The estimate further extends to integral --varifolds of unit density using known regularity and density results.
Paper Structure (6 sections, 19 theorems, 237 equations)

This paper contains 6 sections, 19 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. $W^{1,2}$--Estimate
  3. Controlling the deviation by the CMC defect
  4. Control of the vectorial defect $\vec{H}+2x$
  5. Elliptic upgrade to $W^{2,2}$ and control of the conformal factor
  6. From an area bound to the two--sphere--threshold Willmore bound

Key Result

Theorem 1.1

For any $\alpha\in(0,\tfrac{1}{2})$ there exists $\delta_0=\delta_0(\alpha)>0$ with the following property. Let $F:\Sigma\to\varmathbb{R}^3$ be a smooth immersion of a closed connected surface, let $g:=dF\otimes dF$ and $d\mu_g$ be the induced area measure, and let $\vec{N}$ be a globally defined un Assume Then $\Sigma$ is homeomorphic to $\varmathbb{S}^2$. Moreover, after rescaling so that $\mu_

Theorems & Definitions (35)

  • Theorem 1.1: Qualitative rigidity for almost--CMC spheres BiZhou22
  • Theorem 1.2: Linear quantitative rigidity for almost--CMC spheres
  • Theorem 1.3: Linear quantitative stability under an area bound
  • Theorem 1.4: Varifold stability for almost--CMC spheres
  • Remark 1.5: A finite--perimeter variant
  • Lemma 2.1: Existence of a minimizer
  • proof
  • Lemma 2.2: Orthogonality identities
  • proof
  • Lemma 2.3: Consequences of the normalization
  • ...and 25 more