Coplanar k-Unduloids Are Nondegenerate

Abstract

ABSTRACT. 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 squareintegrable 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.

Geometry and topology of some overdetermined elliptic problems

“…The development of the technique of surface gluing and the construction of constant mean curvature surfaces with Delaunay ends has led to powerful methods in Geometric Analysis, see the paper of N. Kapouleas [22] and the papers of R. Mazzeo, F. Pacard and D. Pollack [26,27]. A related situation is the study of coplanar end surfaces, see the papers by C. Cosin and A. Ros [7], K. Große-Brauckmann, R. Kusner and J. Sullivan [20] and that of these authors joint with N. Korevaar and J. Ratzkin [19]. An other very powerful tool in Geometric Analysis is the use (often for comparison with the maximum principle) of the catenoid, a very well known unbounded minimal surface of revolution.…”

CMC hypersurfaces condensing to Geodesic segments and rays in Riemannian manifolds

“…where Y is the position vector field of k , while the quantities g, N , H, B and g,N ,H ,B have their usual meanings. Since U k is the normal graph of the function r G over the sphere S k as in Section 3, one can replace the integral in (6.4) with an integral over S k , at the expense of an error of size O(ε| log(ε)|r 3 ). Hence by direct computation usingH = 2 r andB i j = rh i j one finds…”

“…Hence to any such surface we may assign a set of parameters describing the asymptotic Delaunay surface corresponding to each end. There are limitations on the asymptotic parameters which can be achieved by complete CMC surfaces, as well as a moduli space theory which describes the infinitesimal and local variations of these asymptotic parameters as one varies the CMC surface, see [3][4][5] and [9].…”

Complete Constant Mean Curvature Hypersurfaces in Euclidean space of dimension four or higher

“…Further progress in this direction was made in [34,35] and also in understanding the moduli space of these surfaces as for example in [37]. Moreover a significant success was that in some cases of genus zero, complete classification results were obtained with a satisfactory understanding of the surfaces involved [9,10,8].…”