$\sigma_k$-Yamabe measure

TL;DR

This paper introduces a divergence structure for the $\sigma_k$-Yamabe operator, deriving a monotonicity formula and interior estimates, leading to the proof of weak continuity of the $\sigma_k$-Yamabe measure.

Contribution

It uncovers a divergence structure for the $\sigma_k$-Yamabe operator and establishes weak continuity of the associated measure, advancing understanding of geometric PDEs.

Findings

Established a divergence structure for the $\sigma_k$-Yamabe operator.

Derived a monotonicity formula for the operator.

Proved weak continuity of the $\sigma_k$-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.