Construction of convex sets in negatively curved manifolds

Borbély, Albert

doi:10.1090/S0002-9939-1993-1166353-0

Construction of convex sets in negatively curved manifolds
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Proc. Amer. Math. Soc. 118 (1993), 205-210 Request permission

Abstract:

It was proved by Choi that one can solve the Dirichlet problem at infinity for simply connected negatively curved manifolds by constructing appropriate convex sets. All the known constructions, it seems, inherently need some kind of growth condition on the curvature; therefore, it is interesting to find new ways to construct convex sets in negatively curved manifolds. In this paper we give a new way to construct convex sets from sets we call $\varepsilon$-almost-convex. From the point of view of this problem this can be considered as a natural generalization of convexity.

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