The curl-Yamabe problem

Abstract

We introduce and study a Yamabe-type variational problem associated with the curl operator on middle-degree forms on a closed oriented Riemannian manifold $(M^n,g)$ with $n\equiv 3(\mathrm{mod}4)$. The corresponding curl--Yamabe constant is defined by minimizing a conformally invariant, gauge-invariant quotient built from $\Vert \mathrm{curl}\alpha {\Vert }_{\frac{2n}{n+1}}$ and ${\int }_M⟨\mathrm{curl}\alpha ,\alpha ⟩$, and its Euler--Lagrange equation is the critical curl--Yamabe equation $\mathrm{curl}\alpha =\frac{n+1}{2}|\alpha {|}^{\frac{2}{n-1}}\alpha$. Using sharp curl--Sobolev inequalities on the sphere and an Aubin-type cut-off construction, we prove that $0<Y_{\mathrm{curl}}(M,[g])\le Y_{\mathrm{curl}}({\mathbb{S}}^n)$ for every such manifold, and that the infimum is attained whenever $Y_{\mathrm{curl}}(M,[g])<Y_{\mathrm{curl}}({\mathbb{S}}^n)$. Moreover, for $n>3$ we establish the strict inequality for manifolds that are not locally conformally flat, via a test-form expansion detecting the Weyl tensor, and deduce existence of a minimizer solving the curl--Yamabe equation.

23 pages