We shall present var­i­ous as­pects of the minimization of func­tion­als for immersion into the spheres of sur­faces under con­strained con­for­mal class. We will focus in par­tic­u­lar on the vari­a­tions of the Will­more and the area func­tional in the class of weak con­for­mal im­mer­sions of given Rie­mann sur­faces. We will in par­tic­u­lar give a char­ac­ter­i­za­tion of rie­mann tori into S3 min­i­miz­ing lo­cally the con­for­mal vol­ume.  

I will present a new compactness theorem for minimal hypersurfaces embedded in a closed Riemannian manifold N^{n+1} with n<7. When n=2 and N has positive Ricci curvature, Choi and Schoen proved that a sequence of minimal hypersurfaces with bounded genus converges smoothly and graphically to some minimal limit. A corollary of our main theorem recovers the result of Choi-Schoen and extends this appropriately for all n<7.