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

Abstract

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 $\int|K(g_k)-1\|d\mu_{g_k}$. We will show that such a sequence is precompact.