The Vacuum Solution

  • Sebastian Klein
Part of the Lecture Notes in Mathematics book series (LNM, volume 2229)


The most obvious solution of the sinh-Gordon equation \(\triangle u + \sinh (u)=0\) is the “vacuum” u = 0, corresponding to a minimal surface of zero sectional curvature, i.e. to a minimal cylinder. It is remarkable that the monodromy and the spectral data can be calculated explicitly for this solution, which we will do in the present chapter. The vacuum solution is of particular importance to us because in the coming chapters, we will describe the asymptotic behavior of the spectral data in the general situation by comparing these data to the corresponding spectral data of the vacuum; it will turn out that the spectral data of any given solution is closely approximated by the spectral data of the vacuum for λ → and for λ → 0.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Sebastian Klein
    • 1
  1. 1.School of Business Informatics & MathematicsUniversity of MannheimMannheimGermany

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