Convergence of manifolds under volume convergence and uniform diameter and tensor bounds

Início: 31/07/2020 17:00

Término: 31/07/2020 18:00

Palestrante: Raquel Perales (Unam)

E-mail do Palestrante:

Resumo: Based on join work with Allen-Sormani and Cabrera Pacheco-Ketterer. Given a Riemannian manifold $M$ and a pair of Riemannian tensors $g_0 leq g_j$ on $M$ it follows that $vol(M)leq vol_j(M)$. Furthermore, the volumes are equal if and only if $g_0=g_j$. In this talk I will show that for a sequence of Riemannian metrics $g_j$ defined on $M$ that satisfy $g_0leq g_j$, $diam (M_j) leq D$ and $vol(M_j)to vol(M_0)$ then $(M,g_j)$ converge to $(M,g_0)$ in the volume preserving intrinsic flat sense. I will present examples demonstrating that under these conditions we do not necessarily obtain smooth, $C^0$ or Gromov-Hausdorff convergence. Furthermore, this result can be applied to show the stability of graphical tori.