Coplanar k-unduloids are nondegenerate

Details Coplanar k-unduloids are nondegenerate Coplanar k-unduloids are nondegenerate

2007-12-12

arxiv.org/abs/0712.1865v2

Int. Math. Res. Not. 2009, 3391-3416

Mathematics

Differential Geometry

19 pages, no figures; improvements to exposition

Scientific paper

We prove each embedded, constant mean curvature (CMC) surface in Euclidean space with genus zero and finitely many coplanar ends is nondegenerate: there is no nontrivial square-integrable solution to the Jacobi equation, the linearization of the CMC condition. This implies that the moduli space of such coplanar surfaces is a real-analytic manifold and that a neighborhood of these in the full CMC moduli space is itself a manifold. Nondegeneracy further implies (infinitesimal and local) rigidity in the sense that the asymptotes map is an analytic immersion on these spaces, and also that the coplanar classifying map is an analytic diffeomorphism.

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