On the nondegeneracy of constant mean curvature surfaces

Abstract

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces $$f:\Sigma \to \mathbb{R}^3$$ are nondegenerate; that is, the Jacobi operator Δ f + | A f |2 has no L2 kernel. In fact, if ∑ has genus zero with k ends, and if f (∑) is embedded (or Alexandrov immersed) in a half-space, then we find an explicit upper bound for the dimension of the L2 kernel 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.