A Cheeger-like inequality for 1-forms on closed 3-manifolds.

It is well known that, on a closed Riemannian manifold, the Laplace operator has discrete spectrum. One can wonder if its first positive eigenvalue has some geometric meaning. Cheeger's seminal work, which is now referred to as Cheeger's inequality, asserts that one can give to this first eigenvalue a geometric lower bound, called the Cheeger's constant. A natural question is to find an analogous statement for differential forms of any degree. During the talk we will review Cheeger's theorem and propose a generalization of Cheeger constant for (coexact) 1-forms on closed 3-manifolds. We shall refer to this constant as the open book constant. If time allows it, we will give some elements of the proof of our main theorem which may be thought as a Cheeger inequality for 1-forms on 3-manifolds: the first eigenvalue of the Hodge Laplacian acting on (coexact) 1-form is bounded from below by the open book constant. This is a joint work with Gilles Courtois (IMJ-PRG, Sorbonne Université).