Spheres of prescribed mean curvature in S^3

We prove a semi-global result on the existence of conformal embeddings of the two-sphere into the round three-sphere S^3(1) with prescribed mean curvature.
Comment: This article is being withdrawn. 1) More general results are now proved in arXiv:1204.5225. 2) As pointed out to me by Jose Espinar (thanks Jose), the area bound proof in reference [11] by Ho is incorrect, which affects the proof of the main result here