John-Nirenberg Radius and Collapse in Conformal Geometry

Given a positive function $u\in W^{1,n}$, we define its John-Nirenberg radius at point $x$ to be the supreme of the radius such that $\int_{B_t}|\nabla\log u|^n<\epsilon_0^n$ when $n>2$, and $\int_{B_t}|\nabla u|^2<\epsilon_0^2$ when $n=2$. We will show that for a collapsing sequence in a fixed conformal class under some curvature conditions, the radius is bounded below by a positive constant. As applications, we will study the convergence of a conformal metric sequence on a $4$-manifold with bounded $\|K\|_{W^{1,2}}$, and prove a generalized H\'elein's Convergence Theorem.