Global geometry of extremal surfaces in three space

e consider the global behavior of complete surfaces in three-space which are extrema for some natural geometric variational problems: surfaces with constant mean curvature H, which extremize Area ;Conservation laws for an elliptic geometric variational problem. We formulate and apply conservation principles for minimal and constant mean curvature surfaces in space forms. We construct on the extremal surface a canonical cohomology moment class with coefficients in the dual Lie algebra of the ambient symmetry group. This leads, for example, to area and curvature estimates for cylindrically-bounded surfaces, and also to balanced geodesic graphs which may serve as moduli for extremal surfaces.;Conformal geometry and complete minimal surfaces. Ideas from conformal geometry and low-dimensional topology are used to study complete minimal surfaces M in R;Comparison surfaces for the Willmore problem. By constructing global comparison surfaces, we estimate the infimum of the conformallly invariant functional W =...