John-Nirenberg Radius and Collapse in Conformal Geometry

Abstract

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 $\Vert K{\Vert }_{W^{1,2}}$, and prove a generalized Hélein's Convergence Theorem.

22 pages

References in corpus (1)

• Conformal metric sequences with integral-bounded scalar curvature

Cited by in corpus (1)

• Compactness of isospectral conformal metrics on 4-manifolds