# The Vacuum Solution

## Abstract

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.