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MiS Preprint
11/2011
A new conformal invariant on 3-dimensional manifolds
Yuxin Ge and Guofang Wang
Abstract
By improving the analysis developed in the study of $\sigma_k$-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 \[ \int_M |Ric-\frac{\overline R } 3 g|^2 dv (g)\le 9\int_M |Ric-\frac{ R } 3 g|^2dv(g), \] where $\overline R=vol (g)^{-1} \int_M R dv(g)$ is the average of the scalar curvature $R$ of $g$. Equality holds if and only if $(M^3,g)$ is a space form. We in fact study the following new conformal invariant \[ \displaystyle \widetilde Y([g_0]):=\sup_{ g\in {\cal C}_1([g_0])}\frac { vol(g)\int_M \sigma_2( g) dv( g)} { (\int_M \sigma_1( g) dv( g))^2}, \] where ${\cal C}_1([g_0]):=\{g=e^{-2u}g_0\,|\, R>0\}$ and prove that $\widetilde Y([g_0])\le 1/3$, which implies the above inequality.
