# Asymptotic Plateau problem in hyperbolic space

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Date/heure
Date(s) - 25/09/2012
11 h 00 min - 12 h 00 min

Catégories Pas de Catégories

$Interior curvature estimates and the asymptotic Plateau problem in Hyperbolic space Joel Spruck In this talk we describe our results on the asymptotic Plateau problem for locally strictly convex hypersurfaces of constant curvature in hyperbolic space Hn+1 which may be formulated as follows: given ? ? ??Hn+1 and a smooth symmetric function f of n variables, we seek a complete locally strictly convex hypersurface ? in Hn+1 satisfying \left(0.1\right) f\left(?\left[?\right]\right) = ? \left(0.2\right) ?? = ? where ?\left[?\right] = \left(?1, . . . , ?n\right) denotes the induced \left(positive\right) hyperbolic principal curva- tures of ? and ? ? \left(0, 1\right) is a constant. The function f is to satisfy the standard structure conditions in the positive cone K=Kn+ :\left[\dots \right]??Rn :eachcomponent?i >0 \left[\dots \right], f > 0 in K, f = 0 on ?K, fi\left(?\right)??f\left(?\right)>0 inK, 1?i?n, ??i \left(0.3\right) \left(0.4\right) \left(0.5\right) \left(0.6\right) \left(0.7\right) In addition, we assume that f is normalized \left(0.8\right) f\left(1,\dots ,1\right) = 1 and satisfies \left(0.9\right) f is homogeneous of degree one. 1 f is a concave function in K.$