Construction of higher genus CMC surfaces in R3 via the generalized Whitham flow

In this thesis, we use the generalized Whitham flow to construct symmetric higher genus g > 1 CMC surfaces in R3. It is well-known that the existence of a conformally immersed CMC surface f : M → R3 is equivalent to the existence of a family of flat connections satisfying the intrinsic (periodicity of the conformal metric) and extrinsic (periodicity of the immersion) closing conditions. The generalized Whitham flow preserves the intrinsic while varies the extrinsic closing conditions. Thereby, the flow parameter, denoted by ρ, determines the genus of the resulting CMC surface. At the initial value ρ = 0 a torus has to be chosen, which in our case is the 3-lobed Wente torus T 2. For tori, the monodromy of the associated family can be parameterized by algebraic data on a hyperelliptic curve, called the spectral curve. The spectral curve of the 3-lobed Wente torus has spectral genus 2 and hyperelliptic reduction allows us to characterize the spectral data in terms of data on elliptic curves. This will help us to derive closing conditions at the initial value ρ = 0. In order to constructed closed symmetric higher genus CMC surfaces in R3, we will study families of flat connections on higher genus Riemann surfaces as the pullback of Fuchsian systems on the 4-punctured sphere, i.e., logarithmic connections on the holomorphically trivial rank two bundle. By that, the underlying Fuchsian system will be parameterized by flat line bundle connections on a torus. This particularly provides useful coordinates to study closing conditions of higher genus CMC surfaces.Investigating the spectral data shows that we have to open two double points outside the unit circle and increase the genus of the spectral curve to 6. By an implicit function theorem argument, we will show that the closing conditions are satisfied for ρ ∈ (−ϵ, ϵ) and prove the existence of compact and branched higher genus CMC surfaces in R3