Optimal rigidity estimates for nearly umbilical surfaces

Let Sigma C R-3 be a smooth compact connected surface without boundary and denote by A its second fundamental form. We prove the existence of a universal constant C such that (1) (inf)(lambda epsilon R) vertical bar vertical bar A - lambda Id vertical bar vertical bar L-(Sigma)(2) <= C vertical bar vertical bar A - (/tr A)(2) id vertical bar vertical bar (2)(L)((Sigma)). Building on this, we also show that, if the right-hand side of (1) is smaller than a geometric constant, Sigma is W-2,W-2 -close to a round sphere.