  The first eigenvalue of the Laplacian on a surface can be viewed as a
functional on the space of Riemannian metrics of a given area. Critical points
of this functional are called extremal metrics. The only known extremal metrics
are a round sphere, a standard projective plane, a Clifford torus and an
equilateral torus. We construct an extremal metric on a Klein bottle. It is a
metric of revolution, admitting a minimal isometric embedding into a 4-sphere
by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for
the Lawson's {3,1}-torus. We conjecture that an extremal metric for the first
eigenvalue on a Klein bottle is unique, and hence it provides a sharp upper
bound for the first eigenvalue on a Klein bottle of a given area. We present
numerical evidence and prove the first results towards this conjecture.
