Constant mean curvature surfaces in metric Lie groups

In these notes we present some aspects of the basic theory on the geometry of a three-dimensional simply-connected Lie group X endowed with a left invariant metric. This material is based upon and extends some of the results of Milnor in Curvatures of left invariant metrics on Lie groups. We then apply this theory to study the geometry of constant mean curvature H >= 0 surfaces in X, which we call H-surfaces. The focus of these results on H-surfaces concerns our joint on going research project with Pablo Mira and Antonio Ros to understand the existence, uniqueness, embeddedness and stability properties of H-spheres in X. To attack these questions we introduce several new concepts such as the H-potential of X, the critical mean curvature 11(X) of X and the notion of an algebraic open book decomposition of X. We apply these concepts to classify the two-dimensional subgroups of X in terms of invariants of its metric Lie algebra, as well as classify the stabilizer subgroup of the isometry group of X at any of its points in terms of these invariants. We also calculate the Cheeger constant for X to be Ch(X) = trace(A), when X = R-2 x(A) R a is a semidirect product for some 2 x 2 real matrix; this result is a special case of a more general theorem by Peyerimhoff and Samiou. We also prove that in this semidirect product case, Ch(X) = 211(X) = 21(X), where I(X) is the infimum of the mean curvatures of isoperimetric surfaces in X. In the last section, we discuss a variety of unsolved problems for H-surfaces in X.