Sharp rigidity estimates for nearly umbilical surfaces

A classical theorem in differential geometry states that if Σ ⊂ R³ is a compact connected surface without boundary and all points of Σ are umbilical, then Σ is a round sphere and therefore its second fundamental form A is a constant multiple of the identity. In a joint work with Stefan Müller we give a sharp quantitative version of this theorem. More precisely we prove the existence of a universal constant C such that $$inflimits_{lambda in } | A - lambda |_{L^2 (Sigma)} leq C | A - frac{tr A}{2} |_{L^2 (Sigma)}$$ Building on this we also show that Σ is W^2,2 close to a round sphere. Both estimates are optimal. Indeed, for p<2, one can exhibit smooth compact connected surfaces with arbitrarly small L^P norm of the traceless part of A and which are close to the union of two distinct round spheres. Our proof uses:

• Ideas of Müller and Sverak to ensure the existence of a conformal parameterization of Σ which enjoys good bounds;
• Codazzi equations and elliptic PDE techniques to estimate the Marcinkievicz norm ∥A - Λ∥_L^2,∞(Σ);
• Elementary algebraic computations combined with Wente-type estimates on skew-symmetric quantities to improve the L^2,∞ bound to the desired L², estimate.