Reference : The Weyl problem for unbounded convex domains in $\HH^3$
 Document type : E-prints/Working papers : Already available on another site Discipline(s) : Physical, chemical, mathematical & earth Sciences : Mathematics To cite this reference: http://hdl.handle.net/10993/47427
 Title : The Weyl problem for unbounded convex domains in $\HH^3$ Language : English Author, co-author : Schlenker, Jean-Marc [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) >] Publication date : 2021 Number of pages : 15 Peer reviewed : No Abstract : [en] Let $K\subset \HH^3$ be a convex subset in $\HH^3$ with smooth, strictly convex boundary. The induced metric on $\partial K$ then has curvature $K>-1$. It was proved by Alexandrov that if $K$ is bounded, then it is uniquely determined by the induced metric on the boundary, and any smooth metric with curvature $K>-1$ can be obtained. We propose here an extension of the existence part of this result to unbounded convex domains in $\HH^3$. The induced metric on $\partial K$ is then clearly not sufficient to determine $K$. However one can consider a richer data on the boundary including the ideal boundary of $K$. Specifically, we consider the data composed of the conformal structure on the boundary of $K$ in the Poincar\'e model of $\HH^3$, together with the induced metric on $\partial K$. We show that a wide range of "reasonable" data of this type, satisfying mild curvature conditions, can be realized on the boundary of a convex subset in $\HH^3$. We do not consider here the uniqueness of a convex subset with given boundary data. Permalink : http://hdl.handle.net/10993/47427 source URL : http://arxiv.org/abs/2106.02101

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