The uniqueness for minimal surfaces inS 3

We give a sufficient condition on a Jordan curve in the 3-dimensional open hemisphereH ofS³ in terms of the Hopf fibering under which spans a unique compact generalized minimal surface inH. The maximum principle for minimal surfaces inS³ is proved and plays an important role in the proof of the uniqueness theorem.Dedicated to Professor Shingo Murakami on his 60th birthdayThis work was carried out while the author was a visitor to the Max-Planck-Institut für Mathematik.