The first eigenvalue of the Laplacian on a surface can be viewedas a functional on the space of Riemannian metrics of a givenarea. Critical points of this functional are called extremalmetrics. The only known extremal metrics are a round sphere, astandard projective plane, a Clifford torus and an equilateraltorus. We construct an extremal metric on a Klein bottle. It is ametric of revolution, admitting a minimal isometric embedding intoa sphere ${mathbb S}^4$ by the first eigenfunctions. Also, thisKlein bottle is a bipolar surface for Lawson's$ au_{3,1}$-torus. We conjecture that an extremal metric for thefirst eigenvalue on a Klein bottle is unique, and hence itprovides a sharp upper bound for $lambda_1$ on a Klein bottle ofa given area. We present numerical evidence and prove the firstresults towards this conjecture.
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