## Journal of Differential Geometry

### Optimal rigidity estimates for nearly umbilical surfaces

#### Abstract

Let $\Sigma \in \mathbf{R}^3$ 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 $$\inf_{\lambda\in {\bf R}}\Vert A - \lambda \rm{Id} \Vert_{L^2(\Sigma)} \leq C \Vert A - \frac{\rm{tr}A}{2} \rm {Id} \Vert_{L^2(\Sigma)^\cdot}$$ Building on this, we also show that, if the right-hand side of (1) is smaller than a geometric constant, Σ is W2,2–close to a round sphere.

#### Article information

Source
J. Differential Geom., Volume 69, Number 1 (2005), 075-110.

Dates
First available in Project Euclid: 16 July 2005

https://projecteuclid.org/euclid.jdg/1121540340

Digital Object Identifier
doi:10.4310/jdg/1121540340

Mathematical Reviews number (MathSciNet)
MR2169583

Zentralblatt MATH identifier
1087.53004

#### Citation

De Lellis, Camillo; Müller, Stefan. Optimal rigidity estimates for nearly umbilical surfaces. J. Differential Geom. 69 (2005), no. 1, 075--110. doi:10.4310/jdg/1121540340. https://projecteuclid.org/euclid.jdg/1121540340