International Mathematics Research Notices Advance Access originally published online on June 5, 2009
International Mathematics Research Notices (2009) 2009:3391-3416, doi:10.1093/imrn/rnp058 published on August 28, 2009
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Coplanar k-Unduloids Are Nondegenerate
1 Technische Universität Darmstadt, Fachbereich Mathematik (AG 3), Schlossgartenstr. 7, 64289 Darmstadt, Germany
2 Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, UT 84112, USA
3 Department of Mathematics, University of Massachusetts, Amherst, MA 01003, USA
4 School of Mathematical Sciences, Aras Na Laoi, University College Cork, Cork, Ireland
5 Technische Universität Berlin, MA 3-2, Straße des 17. Juni 136, 10623 Berlin, Germany
Correspondence: Correspondence to be sent to: sullivan{at}math.tu-berlin.de
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.