Abstract:

I'll describe recent joint work with Xin Zhou on the existence of constant mean curvature 2-spheres in Riemannian 3-spheres. Specifically, we show that in a 3-sphere equipped with an arbitrary Riemannian metric, for almost every H there exists a branched immersed 2-sphere with constant mean curvature H. Moreover, the existence extends to all values of H when the target metric has positive Ricci curvature.