Sharp rigidity estimates for nearly umbilical surfaces

A classical theorem in differential geometry states that if

is a compact connected surface without

boundary and all points of are umbilical, then

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

Building on this we also show that is close to a

round sphere. Both estimates are optimal. Indeed, for p<2,

one can exhibit smooth compact connected surfaces

with arbitrarly small 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 which enjoys good bounds;
• Codazzi equations and elliptic PDE techniques to estimate the Marcinkievicz norm ;
• Elementary algebraic computations combined with Wente-type estimates on skew-symmetric quantities to improve the bound to the desired estimate.