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 $g_{0} l e q g_{j}$ on $M$ it follows that $v o l (M) l e q v o l_{j} (M)$ . 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 $g_{0} l e q g_{j}$ , $d i a m (M_{j}) l e q D$ and $v o l (M_{j}) t o v o l (M_{0})$ 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.