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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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References:
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