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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 \(\mathbb{S}^3\).

53A05 Surfaces in Euclidean and related spaces
46E99 Linear function spaces and their duals
Full Text: DOI
[1] Coifman, R., Lions, P.L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces, J. Math. Pures Appl. 72, 247–286 (1993) · Zbl 0864.42009
[2] De Lellis, C., Müller, S.: Optimal rigidity estimates for nearly umbilical surfaces, to appear in J. Diff. Geom. · Zbl 1087.53004
[3] Fefferman, C., Stein, E.M.: p spaces of several variables, Acta Math. 129, 137–193 (1972) · Zbl 0257.46078 · doi:10.1007/BF02392215
[4] Müller, S., Šverák, V.: On surfaces of finite total curvature, J. Diff. Geom. 42(2), 229–258 (1995) · Zbl 0853.53003
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