On the nondegeneracy of constant mean curvature surfaces

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces $f:\mathrm{\Sigma }\to {\mathbb{R}}^3$ are nondegenerate; that is, the Jacobi operator ${\mathrm{\Delta }}_f+\mathrm{\mid }{A}_f{\mathrm{\mid }}^2$ has no $L^2$ kernel. In fact, if $\mathrm{\Sigma }$ has genus zero and $f(\mathrm{\Sigma })$ is contained in a half-space, then we find an explicit upper bound for the dimension of the $L^2$ jernel in terms of the number of non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields which linearizes the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces to use in gluing constructions, and a surprising characterization of CMC surfaces via spinning spheres.