| Résume | In 1942 P. Alexandrov proved that every Euclidean metric on the 2-sphere with conical singularities of positive curvaturecan be uniquely realized (up to isometry) as the induced metric on the boundary of a convex 3-dimensional 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, Jean-Marc Schlenker proved a similar statement about hyperbolic cusp-metrics on surfaces of genus > 1 (realized in so-called Fuchsian manifolds). Another proof was obtained by François Fillastre. Both of them used the non-constructive "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.
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