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On interior estimates for mean curvature of convex surfaces in \(\mathbb R^3\) and its applications. (English) Zbl 1303.53015

Ge, Molin (ed.) et al., Frontiers in differential geometry, partial differential equations and mathematical physics. In memory of Gu Chaohao. Hackensack, NJ: World Scientific (ISBN 978-981-4578-07-3/hbk; 978-981-4578-08-0/pbk). 105-124 (2014).
Summary: In the present paper a direct proof for the interior estimates for mean curvature of convex surfaces in \(\mathbb R^3\) is given. Meanwhile, when the Gaussian curvature of such a \(C^2\)-convex surface is degenerate on the boundary at degree \(m\) where \(m\) is some integer, a global lower bound away from zero for mean curvature is also obtained. Therefore, global smooth isometric embedding for smooth positive disks in \(\mathbb R^3\) can be constructed by means of the techniques in [J. Hong et al., Commun. Partial Differ. Equations 36, No. 4–6, 635–656 (2011; Zbl 1232.35076)].
For the entire collection see [Zbl 1291.00056].

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Citations:

Zbl 1232.35076
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