Rigidity in the Class of Orientable Compact Surfaces of Minimal Total   Absolute Curvature
  Consider an orientable compact surface in three dimensional Euclidean space with minimum total absolute curvature. If the Gaussian curvature changes sign to finite order and satisfies a nondegeneracy condition along closed asymptotic curves, we show that any other isometric surface differs by at most a Euclidean motion. 
Qing Han, Marcus Khuri
math.DG math.AP