Columnar vortices are stationary solutions of the three-dimensional Euler equations with
axial symmetry, where the velocity field only depends on the distance to the axis and
has no component in the axial direction. Stability of such flows was first investigated
by Lord Kelvin in 1880, but despite a long history the only analytical results available
so far provide necessary conditions for instability under either planar or axisymmetric
perturbations. The purpose of this paper is to show that columnar vortices are spectrally
stable with respect to three-dimensional perturbations with no particular symmetry. Our
result applies to a large family of velocity profiles, including the most common models
in atmospheric flows and engineering applications. The proof is based on a homotopy
argument which allows us, when analyzing the spectrum of the linearized operator, to
concentrate on a small neighborhood of the imaginary axis, where unstable eigenvalues
can be excluded using integral identities and a careful study of the so-called critical layers.
