Circumcenter extension maps for Hadamard manifolds

Abstract:

Given a CAT(-1) space, we can associate to it a boundary at infinity and a cross ratio on said boundary. There is a series of results that tell us that, for sufficiently nice CAT(-1) spaces, the boundary together with the cross ratio uniquely determines the interior space. The proof of these results can be understood in terms of the construction of a circumcenter extension map. It turns out that this relationship between boundary and interior space generalizes nicely to a large class of CAT(0) spaces. In this talk, we will survey some known results, explain the construction of the circumcenter extension and show that the boundary roughly determines the interior space for a large class of manifolds.