John-Nirenberg Radius and Collapse in Conformal Geometry

TL;DR

This paper introduces the John-Nirenberg radius in conformal geometry, demonstrating its lower bound in collapsing sequences and applying it to convergence problems of conformal metrics on 4-manifolds.

Contribution

It defines a new radius concept in conformal geometry and proves its positivity in collapsing sequences, leading to convergence results for conformal metrics.

Findings

John-Nirenberg radius is bounded below in collapsing sequences.

Convergence of conformal metrics on 4-manifolds with bounded curvature is established.

A generalized Hélèin's Convergence Theorem is proved.

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