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 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 Building on this we also show that Σ is W2,2 close to a round sphere. Both estimates are optimal. Indeed, for p<2, one can exhibit smooth compact connected surfaces with arbitrarly small LP 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∥L2,∞(Σ);
  • Elementary algebraic computations combined with Wente-type estimates on skew-symmetric quantities to improve the L2,∞ bound to the desired L2, estimate.

Katharina Matschke

MPI for Mathematics in the Sciences Contact via Mail