Volume Above Distance Below

Abstract

Given a pair of metric tensors g1 ≥ g0 on a Riemannian manifold, M , it is well known that Vol1(M ) ≥ Vol0(M ). Furthermore one has rigidity: the volumes are equal if and only if the metric tensors are the same g1 = g0. Here we prove the that if gj ≥ g0 and Volj(M ) → Vol0(M ) then (M, gj) converge to (M, g0) in the volume preserving intrinsic flat sense. Well known examples demonstrate that one need not obtain smooth, C 0 , Lipschitz, or even Gromov-Hausdorff convergence in this setting. To complete our proof, we provide a new way of estimating the intrinsic flat distance between Riemannian manifolds.

Sobolev Inequalities and Convergence For Riemannian Metrics and Distance Functions

“…Remark 1.4. In the case that p = m 2 we see that the singular set must be Hausdorff dimension 0 which can be seen as a refinement of the main theorem of the first named author, R. Perales, and C. Sormani where volume preserving intrinsic flat convergence is concluded under the same hypotheses [APS20]. In that paper pointwise a.e.…”

“…This shows that there is an analogy for Morrey's inequality for Riemannian manifolds. The work of the first named author, R. Perales, and C. Sormani [APS20] shows that in the ciritcal case where one assumes L m 2 convergence of the Riemannian manifolds, and other geometric conditions, one can obtain Sormani-Wenger Intrinsic Flat convergence of the sequence. In this paper we are interested in studying the sub-critical case p < m 2 where we show a Sobolev inequality between a Riemannian metric and its distance function.…”

Sobolev bounds and convergence of Riemannian manifolds

“…This example is one of a general family of examples which shows that C 0 convergence from below is the right condition to combine with L p convergence (or volume convergence) to imply Gromov-Hausdorff or Sormani-Wenger intrinsic flat convergence. The following theorem is related to the result of Perales, Sormani, and the author [APS20] where if one does not assume a L p 2 bound for p > m then one obtains just volume preserving Sormani-Wenger intrinsic flat convergence of the sequence.…”

“…In both cases, L p bounds on a sequence of metrics was obtained first by studying the scalar curvature PDE for warped products and conformal metrics, respectively. Then, the maximum principle and mean value inequality were used to obtain the necessary C 0 bound from below to apply the main theorem of Sormani and the author in [AS19, AS20] and [APS20], respectively. Hence the theorem of this paper is a type of analogue of those results designed to be applied in a similar way in a case where Gromov-Hausdorff convergence is expected and L p 2 , p > m bounds are naturally obtained or assumed for the metric.…”

“…Review of Volume Above Distance Below. The Perales, Sormani, and the author in [APS20] showed that if one has natural geometric assumptions on a sequence of Riemannian manifolds then one can guarantee volume preserving intrinsic flat convergence to a particular Riemannian manifold. It is by combining the following theorem with Theorem 1.1 which allows us to conclude Theorem 1.4.…”