$\sigma_k$-Yamabe measure

Abstract

We found a special divergence structure for the $\sigma_k$-Yamabe operator and use it to get a monotonicity formula. We also get an interior $L^{\infty}$ estimate via its $L^1$ norm for the $\sigma_k$-Yamabe operator when $1\le k \le \frac{n}{2}$. Combining these two tools, we prove the weak continuity of the $\sigma_k$-Yamabe measure with respect to convergence in measure.