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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.

MSC:
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
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