We relate $L^p$ convergence of metric tensors or volume convergence to a given smooth metric to intrinsic flat and Gromov–Hausdorff convergence for sequences of Riemannian manifolds. We present many examples of sequences of conformal metrics which demonstrate that these notions of convergence do not agree in general even when the sequence is conformal, $g_j=f_j^2{g}_0$, to a fixed manifold. We then prove a theorem demonstrating that when sequences of metric tensors on a fixed manifold $M$ are bounded, $(1-1∕j)g_0\le g_j\le K{g}_0$, and either the volumes converge, ${Vol}_j\left(M\right)\to {Vol}_0\left(M\right)$, or the metric tensors converge in the $L^p$ sense, then the Riemannian manifolds $(M,g_j)$ converge in the measured Gromov–Hausdorff and volume preserving intrinsic flat sense to $(M,g_0)$.

我们迟到了 ${\mathrm{大号}}^p$黎曼流形序列的度量张量收敛或体积收敛到给定的光滑度量到固有平面和Gromov–Hausdorff收敛。我们提供了许多共形指标序列的示例，这些示例证明即使在序列是共形的情况下，这些收敛概念通常也不是一致的，$G_{Ĵ}=F_{Ĵ}^2{G}_0$到固定的歧管。然后我们证明了一个定理，证明了在固定流形上的度量张量序列$\mathrm{中号}$ 有界 $（\mathrm{1个}-\mathrm{1个}∕Ĵ）G_0\le G_{Ĵ}\le ķG_0$，或者两个体积会聚， ${卷}_{Ĵ}（\mathrm{中号}）\to {卷}_0（\mathrm{中号}）$，或度量张量收敛于 ${\mathrm{大号}}^p$ 感，那么黎曼流形 $（\mathrm{中号}，G_{Ĵ}）$ 收敛到测量的Gromov–Hausdorff中，并保留固有的平面感 $（\mathrm{中号}，G_0）$。