Metrics On $S^2$ With Bounded $\mathrm{\parallel }{K}_g{\mathrm{\parallel }}_{L^1\log L^1}$ And Small $\mathrm{\parallel }{K}_g-1{\mathrm{\parallel }}_{L^1}$

In this short paper, we will study the convergence of a metric sequence $g_k$ on $S^2$ with bounded $\int \mathrm{\mid }{K}_{g_k}\mathrm{\mid }\log (1+\mathrm{\mid }{K}_{g_k}\mathrm{\mid })d{\mu }_{g_k}$ and small $\int \mathrm{\mid }K(g_k)-1\mathrm{\parallel }d{\mu }_{g_k}$. We will show that such a sequence is precompact.