A C 0 Estimate for Nearly Umbilical Surfaces

Let Σ ⊂ R ³ be a smooth compact connected surface without boundary. Denote by A its second fundamental form and by Å the tensor A−(tr A/2)Id. In 4] we proved that, if ‖Å‖ _L ² (Σ) is small, then Σ is W ^2,2-close to a round sphere. In this note we show that, in addition, the metric of Σ is C ⁰–close to the standard metric of S ².The first author was supported by a grant of the Swiss National Science Foundation.