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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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##### References:
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