Sharp rigidity estimates for nearly umbilical surfaces

  • Camillo DeLellis (MPI MiS, Leipzig)
A3 01 (Sophus-Lie room)


A classical theorem in differential geometry states that if

formula26 is a compact connected surface without

boundary and all points of formula28 are umbilical, then

formula28 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 formula28 is formula38 close to a

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

one can exhibit smooth compact connected surfaces

with arbitrarly small formula42 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 formula28 which enjoys good bounds;
  • Codazzi equations and elliptic PDE techniques to estimate the Marcinkievicz norm formula48;
  • Elementary algebraic computations combined with Wente-type estimates on skew-symmetric quantities to improve the formula50 bound to the desired formula52 estimate.

Katharina Matschke

MPI for Mathematics in the Sciences Contact via Mail