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

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.

MSC:

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