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Foliations of hyperbolic space by constant mean curvature surfaces sharing ideal boundary. (English) Zbl 1081.53029
A Jordan curve \(\gamma\) in a unit sphere \(S^2\), considered as the ideal boundary of the hyperbolic space \(\mathbb H^3\) of curvature 1, divides \(S^2\) into two spherical caps \(G_{\pm1}\) that present simple connected discs of constant mean curvature \(\pm1\). It is known [B. Guan and J. Spruck, Am. J. Math. 122, No. 5, 1039–1060 (2000; Zbl 0990.53055)] that under certain circumstances for any \(c\in(-1,1)\) there is a disc of constant mean curvature \(c\) embedded in \(\mathbb H^3\) with \(\gamma\) as its ideal boundary. Using analysis and numerical experiments (the level set method, [S. Osher and J. A. Sethian, J. Comput. Phys. 79, No. 1, 12–49 (1988; Zbl 0659.65132)], to compute the flow), the authors examine whether or not these surfaces (presented by level sets of some regular function \(F:\mathbb H^3\to\mathbb R\) that defines a parabolic flow) in fact foliate \(\mathbb H^3\), and to what extent the known conditions on the curve can be relaxed.

53C12 Foliations (differential geometric aspects)
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
Full Text: DOI EuDML
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