Curvature estimates for surfaces with bounded mean curvature

Bourni, Theodora; Tinaglia, Giuseppe

doi:10.1090/S0002-9947-2012-05487-0

Curvature estimates for surfaces with bounded mean curvature
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by Theodora Bourni and Giuseppe Tinaglia PDF

Trans. Amer. Math. Soc. 364 (2012), 5813-5828 Request permission

Abstract:

Estimates for the norm of the second fundamental form, $|A|$, play a crucial role in studying the geometry of surfaces in $\mathbb {R}^3$. In fact, when $|A|$ is bounded the surface cannot bend too sharply. In this paper we prove that for an embedded geodesic disk with bounded $L^2$ norm of $|A|$, $|A|$ is bounded at interior points, provided that the $W^{1,p}$ norm of its mean curvature is sufficiently small, $p>2$. In doing this we generalize some renowned estimates on $|A|$ for minimal surfaces.

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