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Geodesics and soap bubbles in surfaces. (English) Zbl 0865.53009

On a Riemannian surface there is a curve, often of constant curvature, which minimizes length among all curves bounding the same area or curvature. This curve can be required to satisfy additional topological restrictions. One consequence is a simple proof of an argument, suggested by Poincaré, that the shortest curve which divides a convex 2-sphere into two regions of total curvature \(2\pi\) is a simple geodesic.

MSC:

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

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