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 $n\equiv3\pmod4$. The corresponding curl--Yamabe constant is defined by minimizing a conformally invariant, gauge-invariant quotient built from $\|{\rm curl}\,\alpha\|_{\frac{2n}{n+1}}$ and $\int_M\langle{\rm curl}\,\alpha,\alpha\rangle$, and its Euler--Lagrange equation is the critical curl--Yamabe equation ${\rm curl}\,\alpha=\frac{n+1}{2}|\alpha|^{\frac{2}{n-1}}\alpha$. Using sharp curl--Sobolev inequalities on the sphere and an Aubin-type cut-off construction, we prove that $0<Y_{\rm curl}(M,[g])\le Y_{\rm curl}(\mathbb{S}^n)$ for every such manifold, and that the infimum is attained whenever $Y_{\rm curl}(M,[g])<Y_{\rm curl}(\mathbb{S}^n)$. 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.