Metrics On $S^2$ With Bounded $\Vert K_g{\Vert }_{L^1\log L^1}$ And Small $\Vert K_g-1{\Vert }_{L^1}$

Yuxiang Li, Hongyan Tang

In this short paper, we will study the convergence of a metric sequence $g_k$ on $S^2$ with bounded $\int |K_{g_k}|\log (1+|K_{g_k}|)d{\mu }_{g_k}$ and small $\int |K(g_k)-1\Vert d{\mu }_{g_k}$. We…