On the nondegeneracy of constant mean curvature surfaces

Details On the nondegeneracy of constant mean curvature surfaces On the nondegeneracy of constant mean curvature surfaces

2004-07-09

arxiv.org/abs/math/0407153v3

Geom. Funct. Anal. 16 (2006), 891-923

Mathematics

Differential Geometry

v2: substantial revisions, to appear in Geom. Funct. Anal.; three figures

Scientific paper

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces $f:\Sigma \to \R^3$ are nondegenerate; that is, the Jacobi operator $\Delta_f + |A_f|^2$ has no $L^2$ kernel. In fact, if $\Sigma$ has genus zero and $f(\Sigma)$ 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.

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