Infinitesimal Darboux transformations of the spectral curves of tori in the four-space. (English) Zbl 1141.53009

Grinevich, P. G.; Taimanov, I. A.

Int. Math. Res. Not. 2007, No. 2, Article ID rnm005, 21 p. (2007).

This paper is partly a survey of spectral curves techniques for surfaces in \(\mathbb{R}^3\) and \(\mathbb{R}^4\), while bringing in new results on the spectral curve of a torus in \(\mathbb{R}^4\) under conformal transformations of \(\overline{\mathbb{R}}^4\). The motivation for using these techniques lies in the fact that spectral curves relate significantly to the geometry of surfaces. In fact, the second author proposed proving Willmore’s conjecture using spectral curves.
The authors show here that any conformal transformation of \(\overline{\mathbb{R}}^4\) which maps a torus in \(\mathbb{R}^4\) into a compact surface preserves the multiplier set of the corresponding Dirac operator.

Reviewer: Alina Stancu (Lowell)

MSC:

53A30	Conformal differential geometry (MSC2010)
37K15	Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems

Keywords:

conformal transformations; Dirac operator; Floquet multipliers; Weierstrass representation