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A \(C^{0}\) estimate for nearly umbilical surfaces. (English) Zbl 1100.53005
If, for a smooth compact connected surface in \(\mathbb{R}^3\), the traceless part of the second fundamental form is small in the \(L^2\)-norm, then it is known that the surface is \(W^{2,2}\)-close to a round sphere. In the present paper it is shown that in addition the surface metric is \(C^0\)-close to the standard metric of \(\mathbb{S}^2\). The proof makes use of a Hardy bound and the Hopf fibration of \(\mathbb{S}^3\).

MSC:
53A05 Surfaces in Euclidean and related spaces
46E99 Linear function spaces and their duals
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