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John–Nirenberg radius and collapsing in conformal geometry
Asian Journal of Mathematics ( IF 0.6 ) Pub Date : 2020-10-01 , DOI: 10.4310/ajm.2020.v24.n5.a2 Yuxiang Li 1 , Guodong Wei 2 , Zhipeng Zhou 3
Asian Journal of Mathematics ( IF 0.6 ) Pub Date : 2020-10-01 , DOI: 10.4310/ajm.2020.v24.n5.a2 Yuxiang Li 1 , Guodong Wei 2 , Zhipeng Zhou 3
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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.
中文翻译:
约翰·尼伦贝格半径和共形几何体坍塌
给定一个正函数$ u \ in W ^ {1,n} $,我们将其在点$ x $处的John–Nirenberg半径定义为半径的最大值,使得$ \ int_ {B_t(x)} {\ lvert \ nabla \ operatorname {log} u \ rvert} ^ n \ lt \ epsilon ^ n_0 $当$ n \ gt 2 $和$ \ int_ {B_t(x)} {\ lvert \ nabla u \ rvert} ^ 2 \ lt \ epsilon ^ 2_0 $,当$ n = 2 $时。我们将显示,对于在某些曲率条件下固定的共形分类中的一系列崩溃指标,半径以正常数为界。作为应用程序,我们将研究带约束的$ {\ lVert K \ rVert} _ {W ^ {1,2}} $在$ 4 $流形上的共形度量序列的收敛性,并证明广义Hélein的收敛定理。
更新日期:2020-10-01
中文翻译:
约翰·尼伦贝格半径和共形几何体坍塌
给定一个正函数$ u \ in W ^ {1,n} $,我们将其在点$ x $处的John–Nirenberg半径定义为半径的最大值,使得$ \ int_ {B_t(x)} {\ lvert \ nabla \ operatorname {log} u \ rvert} ^ n \ lt \ epsilon ^ n_0 $当$ n \ gt 2 $和$ \ int_ {B_t(x)} {\ lvert \ nabla u \ rvert} ^ 2 \ lt \ epsilon ^ 2_0 $,当$ n = 2 $时。我们将显示,对于在某些曲率条件下固定的共形分类中的一系列崩溃指标,半径以正常数为界。作为应用程序,我们将研究带约束的$ {\ lVert K \ rVert} _ {W ^ {1,2}} $在$ 4 $流形上的共形度量序列的收敛性,并证明广义Hélein的收敛定理。