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Zbl 0578.49027
Anderson, Michael T.
Curvature estimates for minimal surfaces in 3-manifolds. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 18, 89-105 (1985). ISSN 0012-9593

The main result of this paper is an interior curvature estimate for minimal embedded discs in 3-manifolds, which is obtained without stability assumptions. The general approach used is to first prove a global result and to then deduce from it the local curvature estimate. In the case of the main theorem mentioned above, the global result is a characterization of the plane in ${\bbfR}\sp 3$ as the only complete embedded minimal surface of finite topological type with one end which is of quadratic area growth. The author also obtains a curvature estimate at the boundary for embedded minimal discs and estimates for immersed minimal discs with boundary of total curvature strictly less than $6\pi$. The proofs, besides involving differential geometry, rely on compactness results for varifolds and monotonicity for stationary varifolds.
[H.Parks]

MSC 2000:: *49Q05 Minimal surfaces (calculus of variations)
49Q15 Geometric measure and integration theory, etc.
53A10 Minimal surfaces, surfaces with prescribed mean curvature
53C42 Immersions (differential geometry)

Keywords: interior curvature estimate; minimal embedded discs; 3-manifolds; stationary varifolds

Cited in: Zbl 0880.53033

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