Abstract
We give a complete description of the flat surfaces in hyperbolic 3-space that are regularly embedded around an isolated singularity. Specifically, we show that there is a one-to-one explicit correspondence between this class and the class of regular analytic convex Jordan curves in the 2-sphere. Previously, the only known examples of such surfaces were rotational ones. To achieve this result, we first solve the geometric Cauchy problem for flat surfaces in hyperbolic 3-space.
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