A new conformal invariant on 3-dimensional manifolds

By improving the analysis developed in the study of ${\text{s}}_k$-Yamabe problem, we prove in this paper that the De Lellis-Topping inequality is true on 3-dimensional Riemannian manifolds of nonnegative scalar curvature. More precisely, if $(M^3,g)$ is a 3-dimensional closed Riemannian manifold with non-negative scalar curvature, then $${\int }_M\mathrm{\mid }Ric-\frac{\bar{R}}{3}g{\mathrm{\mid }}^{2}dv(g)\le 9{\int }_M\mathrm{\mid }Ric-\frac{R}{3}g{\mathrm{\mid }}^{2}dv(g),$$ where $\bar R=vol (g)^{-1} \int_M R dv(g)$istheaverageofthescalarcurvature$R$of$g$. Equality holds if and only if $(M^3,g)$ is a space form. We in fact study the following new conformal invariant $$\text{ds}\tilde{Y}([g_0]):=\underset{g\in {\mathcal{C}}_1([g_0])}{\sup }\frac{\text{ds}vol(g){\int }_M{\text{s}}_2(g)dv(g)}{\text{ds}({\int }_M{\text{s}}_1(g)dv(g){)}^2},$$ where ${\mathcal{C}}_1([g_0]):=\{g=e^{-2u}{g}_0\mathrm{\mid }R>0\}$ and prove that $\tilde{Y}([g_0])\le 1\mathrm{/}3$, which implies the above inequality.