Optimal metrics for the first curl eigenvalue on 3-manifolds

In this article we analyze the spectral properties of the curl operator on closed Riemannian 3-manifolds. Specifically, we study metrics that are optimal in the sense that they minimize the first curl eigenvalue among any other metric of the same volume in the same conformal class. We establish a connection between optimal metrics and the existence of minimizers for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi> <mml:mfrac><mml:mn>3</mml:mn> <mml:mn>2</mml:mn></mml:mfrac> </mml:msup> </mml:math> -norm in a fixed helicity class, which is exploited to obtain necessary and sufficient conditions for a metric to be locally optimal. As a consequence, our main result is that we prove that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow> <mml:mn>3</mml:mn></mml:msup> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi> <mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow> <mml:mn>3</mml:mn></mml:msup> </mml:mrow> </mml:math> endowed with the round metric are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>C</mml:mi> <mml:mn>1</mml:mn></mml:msup> </mml:math> -local minimizers for the first curl eigenvalue (in its conformal and volume class). The connection between the curl operator and the Hodge Laplacian allows us to infer that the canonical metrics of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow> <mml:mn>3</mml:mn></mml:msup> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi> <mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow> <mml:mn>3</mml:mn></mml:msup> </mml:mrow> </mml:math> are locally optimal for the first eigenvalue of the Hodge Laplacian on coexact 1-forms. This is in strong contrast to what happens in four dimensions.