Coplanar k-unduloids are nondegenerate

TL;DR

This paper proves that embedded, genus-zero, coplanar constant mean curvature surfaces are nondegenerate, ensuring their moduli space forms a smooth real-analytic manifold and establishing their local rigidity and classification.

Contribution

It demonstrates the nondegeneracy of coplanar k-unduloids, leading to a smooth moduli space and a precise classification of these surfaces.

Findings

Moduli space of coplanar CMC surfaces is a real-analytic manifold.

Coplanar surfaces exhibit local rigidity and are classified by an analytic diffeomorphism.

No nontrivial square-integrable solutions to the Jacobi equation for these surfaces.

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 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.