Quantitative Stability of the Clifford Torus as a Willmore Minimizer

TL;DR

This paper proves that surfaces with nearly minimal Willmore energy in the 3-sphere are quantitatively close to the Clifford torus after conformal normalization, demonstrating stability of the Clifford torus as a minimizer.

Contribution

It establishes a quantitative stability result for the Clifford torus as a minimizer of the Willmore energy among genus-one surfaces in the 3-sphere.

Findings

Surfaces with Willmore energy close to 2π² are close to the Clifford torus after conformal transformation.

The deviation in conformal factor and metric coefficients is proportional to the energy excess.

The result provides a stability estimate in the W^{2,2} norm for the conformal parametrization.

Abstract

For an integral $2$-varifold $V\subset \mathbb{S}^3$ with square-integrable mean curvature, unit density, and support of genus at least $1$, assume that its Willmore energy satisfies \[ \mathcal{W}(V)\le 2\pi^2+\delta^2,\qquad \delta<\delta_0\ll1. \] We show that the support $\Sigma=\operatorname{spt}V$ is, after applying a suitable conformal transformation of $\mathbb{S}^3$, quantitatively close to the Clifford torus. More precisely, under an appropriate conformal normalization, the surface $\Sigma$ admits a $W^{2,2}$ conformal parametrization by the flat torus whose conformal factor and metric coefficients differ from those of the Clifford torus by at most $C\delta$.