Greatest least eigenvalue of the Laplacian on the Klein bottle

TL;DR

This paper proves a conjecture relating the least eigenvalue of the Laplacian and the area of the Klein bottle, identifying the extremal metric and analyzing an integrable Hamiltonian system.

Contribution

It establishes the sharp upper bound for the eigenvalue-area product on the Klein bottle and characterizes the unique extremal metric achieving equality.

Findings

Proved the eigenvalue-area inequality for the Klein bottle.

Identified the unique extremal metric up to dilations.

Analyzed a completely integrable Hamiltonian system related to the problem.

Abstract

We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: For any Riemannian metric $g$ on the Klein bottle $\mathbb{K}$ one has $$\lambda\_1 (\mathbb{K}, g) A (\mathbb{K}, g)\le 12 \pi E(2\sqrt 2/3),$$ where $\lambda\_1(\mathbb{K},g)$ and $A(\mathbb{K},g)$ stand for the least positive eigenvalue of the Laplacian and the area of $(\mathbb{K},g)$, respectively, and $E$ is the complete elliptic integral of the second kind. Moreover, the equality is uniquely achieved, up to dilatations, by the metric $$g\_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos^2v} (du^2 + {dv^2\over 1+8\cos ^2v}),$$ with $0\le u,v <\pi$. The proof of this theorem leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures.