Résume  In 1942 P. Alexandrov proved that every Euclidean metric on the 2sphere with conical singularities of positive curvaturecan be uniquely realized (up to isometry) as the induced metric on the boundary of a convex 3dimensional polytope.
It provided a complete inner description of such metrics and was used in the development of a general theory of metrics with nonnegative curvature.
Various authors gave several generalizations of this result. In particular, JeanMarc Schlenker proved a similar statement about hyperbolic cuspmetrics on surfaces of genus > 1 (realized in socalled Fuchsian manifolds). Another proof was obtained by François Fillastre. Both of them used the nonconstructive "deformation method".
In our talk we will discuss a variational approach to this problem. We will mention the relation with discrete uniformization theory and consider possible perspectives of our technique.
