# John-Nirenberg Inequality and Collapse in Conformal Geometry

Abstract: Let $g$ be a metric over $B$ which is conformal to $g_0$.We assume $\|R(g_k)\|_{L^p} <C$, where $R$ is the scalar curvature and $p\geq \frac{n}{2}$.We will use the John-Nirenberg inequality to prove that if $vol(B,g_k) \rightarrow 0$, then there exists $c_k \rightarrow+\infty$, such that $c_ku_k$
converges to a positive function weakly in $W^{2,p}_{loc}(B)$. As an application, we will study the bubble tree convergence of a conformal metric sequence with integral-bounded scalar curvature.

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