In this talk, I will describe the regularity theory for surfaces minimizing the prescribed mean curvature functional over isotopies in a closed Riemannian 3-manifold, which is a prescribed mean curvature counterpart of the celebrated regularity result of Meeks, Simon and Yau about minimizers of the area functional over isotopies. 

Whereas for the area functional minimizers over isotopies are smooth embedded minimal surfaces, minimizers of the prescribed mean curvature functional turn out to be C^{1,1} immersions which can have a large self-touching set where the mean curvature vanishes. 

Even though the proof broadly follows the same general strategy as in the case of the area functional, several new ideas are needed to deal with the lower regularity setting. This regularity theory plays an important role in Z. Wang-X. Zhou’s recent proof of the existence of 4 embedded minimal spheres in a generic metric on the 3-sphere.

The results in this talk are joint work with Douglas Stryker (Princeton).