Moduli Space Theory for Constant Mean Curvature Surfaces Immersed in Space-forms

The study of constant mean curvature surfaces in a space-form has been an active field since the work of H. Hopf in the 1920’s and H.Liebmann in the years around 1900. The questions which are generally of interest are global questions of existence and uniqueness in complete 3-manifolds. We deal in this short paper on a question of existence and uniqueness with respect to the complex structure and the quadratic Hopf differential of a compact surface in a constant curvature 3-manifold which is not necessarily complete.Our final result applies only to the case of surfaces embedded in a local 3-dimensional space of constant curvature −1, where the mean curvature constant c satisfies |c| < 1. However, the technique suggests some approaches to the more interesting cases, for example the work of Bryant [B] and Kenmotsu [K]. The Gauss-Codazzi equations for constant mean curvature immersions of a surface into a 3-dimensional space-form are a 3 × 3 system of partial differential equations of mixed order. Once a complex structure is chosen, the equations break down into two equations. The Codazzi equation on the second fundamental form yields the CauchyRiemann equation for a holomorphic quadratic differential first noticed and used by Heintz Hopf [H]. The second is a real non-linear single elliptic equation for the length function of the metric which comes from the Gauss curvature equation. These equations can be approached via a number of techniques in partial differential equations. In this short note, we improve upon results obtained by assuming the Riemann surface structure and postulating a fixed quadratic differential representing the (2, 0) part of a second fundamental form as solving the Codazzi equations. This leaves the problem of solving the elliptic Gauss equation for the length function of the metric. By analyzing the Gauss and Codazzi equations together, we are able to reformulate the equations in a form which completely identifies all local solutions in the case of negative curvature. We prove that the moduli space of solutions to the Gauss-Codazzi equations for a Riemann surface of genus greater than one immersed with mean curvature constant and less than 1, in a not necessarily complete 3-manifold of constant curvature −1 is parameterized by cohomology classes of (0, 2) differentials. The result is similar and proved in the same fashion as the results in gauge theory in a paper of the first author [G]. In fact, the details of how the computations change with the change in base-point g is comlicated, but it is familiar to geometers from the variational formulation of the Yamabe problem and will not be repeated. An abstract proof could be constructed along the lines of the convexity theory used in the gauge theory