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BIRS Workshop Lecture Videos

Convergence of manifolds under volume convergence, a tensor and a diameter bound Perales, Raquel


Given a closed and oriented manifold $M$ and Riemannian tensors $g_0 \leq g_j$ on $M$ that satisfy $vol(M, g_j)\to vol(M,g_0)$ and $diam(M,g_j)\leq D$ we will see that $(M,g_j)$ converges to $(M,g_0)$ in the volume preserving intrinsic flat sense. We note that under these conditions we do not necessarily obtain smooth, $C^0$ or even Gromov-Hausdorff convergence. Nonetheless, this result can be applied to show stability of a class of tori. That is, any sequence of tori in this class with almost nonnegative scalar curvature converge to a flat torus. We will also see that an analogous convergence result to the stated above but for manifolds with boundary can be applied to show stability of the positive mass theorem for a particular class of manifolds. [Based on joint works with Allen, Allen-Sormani, Cabrera Pacheco - Ketterer, and Huang - Lee]

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