Singular curves and the Plateau problem. (English) Zbl 0737.49029

Hass, Joel

Int. J. Math. 2, No. 1, 1-16 (1991).

T. Radó [Math. Z. 32, 763-795 (1930; JFM 56.0436.01)] and J. Douglas [Trans. Am. Math. Soc. 33, 263-321 (1931; Zbl 0001.14102)] showed that an embedded rectifiable curve bounds a disk of least area. The author shows that this result holds for all rectifiable curves, including singular ones. More precisely he proves the following theorem: “Let \(\gamma\) be a rectifiable curve in \(R^ n\). Then \(\gamma\) bounds a disk of least area, which is a smooth branched immersion away from the boundary. If \(n=3\), then \(\gamma\) bounds a disk of least area which is a smooth immersion away from the boundary”. At the end the author extends this theorem to the general Riemannian manifold setting.

Reviewer: C.Udrişte (Bucureşti)

MSC:

49Q05	Minimal surfaces and optimization
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

embedded rectifiable curve; disk of least area

Citations:

Zbl 0001.14102; JFM 56.0436.01