John–Nirenberg radius and collapsing 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 $\int_{B_t (x)} {\lvert \nabla \operatorname{log} u \rvert}^n \lt \epsilon^n_0$ when $n \gt 2$, and $\int_{B_t (x)} {\lvert \nabla u \rvert}^2 \lt \epsilon^2_0$ when $n = 2$. We will show that for a collapsing sequence of metrics 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 ${\lVert K \rVert}_{W^{1,2}}$, and prove a generalized Hélein’s Convergence Theorem.

Keywords

John–Nirenberg radius, scalar curvature equation, blow-up analysis

2010 Mathematics Subject Classification

53C21, 58J05

Full Text (PDF format)

Received 17 April 2018

Accepted 22 January 2020

Published 10 March 2021