I will present some of the basic results on the geometry of constant mean curvature H≥0 surfaces M in a complete homogeneous 3-manifold X; such an M will be called an H-surface. We will call a foliation F of X a CMC foliation if all of the leaves of F are H-surfaces with H possibly varying. The key results in the mini-course will include:

• General theory of H-surfaces M in Riemannian 3-manifolds with an emphasis on the case M is complete and embedded, including work described in papers of Colding-Minicozzi and Meeks-Perez-Ros and Meeks-Tinaglia.
• Curvature estimates for CMC foliations of X. Based on joint work with Perez and Ros.
• Curvature estimates for H-disks with H>0 in X. Based on joint work with Tinaglia.
• Uniqueness results for H-spheres in X, which generalize the classical result of Hopf that for each H>0, there is a unique H-sphere in R³. Based on joint work with Mira, Perez and Ros.