Asian Journal of Mathematics

Volume 24 (2020)

Number 5

John–Nirenberg radius and collapsing in conformal geometry

Pages: 759 – 782

DOI: https://dx.doi.org/10.4310/AJM.2020.v24.n5.a2

Authors

Yuxiang Li (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Guodong Wei (School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, Guangdong, China)

Zhipeng Zhou (Academy of Mathematics and Systems Science, CAS, Beijing, China)

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 $\int_{B_t (x)} {\lvert \nabla \operatorname{log} u \rvert}^n \lt \epsilon^n_0$ 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

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Received 17 April 2018

Accepted 22 January 2020

Published 10 March 2021