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Parametric surfaces of prescribed mean curvature. (English) Zbl 0955.53033

Hildebrandt, S. (ed.) et al., Calculus of variations and geometric evolution problems. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (CIME), Cetraro, Italy, June 15-22, 1996. Berlin: Springer. Lect. Notes Math. 1713, 211-265 (1999).
The paper under review contains lectures given by the author on the Plateau problem with prescribed mean curvature \(H\), where \(H\) is a given real function on \(\mathbb{R} ^3\). The problem is to find a surface \(S\) with prescribed boundary curve \(\Gamma\) and given mean curvature \(H(x)\) at each point \(x\) of \(S\). This problem is a generalization of the classical Plateau problem where one looks for area-minimizing surfaces \(S\) spanning a given boundary curve \(\Gamma\). The author starts with a presentation of the theory for parametric surfaces with prescribed mean curvature in \(\mathbb{R} ^3\), which originated in the early 70’s. Then he describes the geometric measure theory approach to the problem for hypersurfaces in \(\mathbb{R} ^{n+1}\) and finishes with a discussion of ideas for the very recent and still in progress results, in the case where the ambient space is a Riemannian manifold.
For the entire collection see [Zbl 0927.00029].

MSC:

53C40 Global submanifolds
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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