[Paper Review] On infinitesimal deformations of cmc surfaces of finite type in the 3-sphere

Martin Kilian, Martin Schmidt|ArXiv.org|Oct 12, 2009

Geometric Analysis and Curvature Flows10 references3 citations

TL;DR

This paper studies infinitesimal deformations of constant mean curvature (CMC) surfaces of finite type in the 3-sphere using integrable systems techniques. It employs Baker-Akhiezer functions, polynomial Killing fields, and spectral curves to classify deformations as isospectral or non-isospectral, showing that non-isospectral deformations in S³ can change the mean curvature—unlike in R³—while proving the space of isospectral deformations is g-dimensional, where g is the genus of the spectral curve.

ABSTRACT

We describe infinitesimal deformations of constant mean curvature surfaces of finite type in the 3-sphere. We use Baker-Akhiezer functions to describe such deformations, as well as polynomial Killing fields and the corresponding spectral curve to distinguish between isospectral and non-isospectral deformations.

Motivation & Objective

• To extend the theory of infinitesimal deformations of CMC tori from R³ to S³, where mean curvature can change under deformation.
• To characterize infinitesimal deformations of CMC surfaces of finite type in S³ using integrable systems tools such as Baker-Akhiezer functions and polynomial Killing fields.
• To distinguish between isospectral and non-isospectral deformations via the spectral curve of the sinh-Gordon equation.
• To show that the space of isospectral deformations is g-dimensional, with g the arithmetic genus of the spectral curve.
• To construct non-isospectral deformations that alter the mean curvature by modifying the spectral curve's branch points.

Proposed method

• Uses the extended frame formalism and the Maurer–Cartan equation to describe conformal immersions into S³ via SU(2)-valued maps.
• Applies the Dorfmeister–Pedit–Wu construction to represent CMC surfaces via solutions of the sinh-Gordon equation.
• Introduces polynomial Killing fields and their associated spectral curves to classify deformations of CMC surfaces.
• Constructs Jacobi fields and parametric Jacobi fields using the Baker-Akhiezer function of the sinh-Gordon equation.
• Derives inhomogeneous Jacobi equations for normal variations under non-isospectral deformations.
• Uses the Fermi curve of the Jacobi operator to show isomorphism with the spectral curve of the sinh-Gordon equation.

Experimental results

Research questions

• RQ1 How do infinitesimal deformations of CMC tori in S³ differ from those in R³, particularly regarding mean curvature changes?
• RQ2 What role do Baker-Akhiezer functions play in constructing Jacobi fields and parametric deformations for CMC surfaces in S³?
• RQ3 How can polynomial Killing fields be used to distinguish between isospectral and non-isospectral deformations?
• RQ4 What is the dimension of the space of isospectral deformations, and how is it related to the spectral curve’s genus?
• RQ5 Can non-isospectral deformations be explicitly constructed via modifications of the spectral curve’s branch points?

Key findings

• Non-isospectral deformations of CMC tori in S³ exist and can change the mean curvature, unlike in R³ where such deformations are isospectral.
• The space of isospectral deformations is g-dimensional, where g is the arithmetic genus of the spectral curve, and is generated by linear combinations of transformations at branch points.
• The Fermi curve of the Jacobi operator is isomorphic to the spectral curve of the sinh-Gordon equation, establishing a deep link between deformation theory and spectral data.
• Infinitesimal non-isospectral deformations are constructed by modifying the derivative of the bilinear form at a branch point a, with the deformation vector field proportional to the commutator involving Q and ξ.
• The deformation vector field at a branch point a is given by (λ−a)⁻¹ times a commutator expression involving Q and diagonal matrices, and this generates all non-isospectral deformations.
• All non-trivial holomorphic one-forms on the spectral curve yield non-trivial relations among infinitesimal isospectral transformations, confirming the g-dimensional nature of the isospectral space.