A $W^{1,p}$-metric on an n-dimensional closed Riemannian manifold naturally induces a distance function, provided p is sufficiently close to n. If a sequence of metrics $g_k$ converges in $W^{1,p}$ to a limit metric $g$, then the corresponding distance functions $d_{g_k}$ subconverge to a limit distance function d, which satisfies $d\le d_g$.

As an application, we show that the above convergence result applies to a sequence of conformal metrics with $L^{n/2}$-bounded scalar curvatures, under certain geometric assumptions. In particular, in this special setting, the limit distance function d actually coincides with $d_g$.

中文翻译：

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W1、p-度量和具有 Ln/2 有界标量曲率的共形度量

一个$W^{\mathrm{1个},p}$n维闭黎曼流形上的 -metric自然会导出距离函数，前提是p足够接近n。如果一系列指标$G_k$收敛于$W^{\mathrm{1个},p}$到极限指标$G$, 那么相应的距离函数$d_{G_k}$子收敛到极限距离函数d，它满足$d\le d_G$.

作为一个应用，我们证明了上述收敛结果适用于一系列具有${\mathrm{大号}}^{n/\mathrm{2个}}$- 有界标量曲率，在某些几何假设下。特别地，在这种特殊设置下，极限距离函数d实际上与$d_G$.