Inverse spectral theory and the Minkowski problem for the surface of   revolution
  We solve the inverse spectral problem for rotationally symmetric manifolds, which include the class of surfaces of revolution, by giving an analytic isomorphism from the space of spectral data onto the space of functions describing the radius of rotation. An analogue of the Minkowski problem is also solved. 
Hiroshi Isozaki, Evgeny L. Korotyaev
math-ph math.MP math.SP