Geometric Convergence to an Extreme Limit Space with nonnegative scalar curvature

TL;DR

This paper constructs a sequence of nonnegatively curved manifolds converging to an extreme limit space, providing a test case for generalized scalar curvature concepts in geometric analysis.

Contribution

It introduces a new extreme limit space with nonnegative scalar curvature as a convergence target for warped product manifolds, advancing the understanding of scalar curvature limits.

Findings

Sequence converges in volume preserving intrinsic flat and Gromov-Hausdorff senses.

Limit space features warping function hitting infinity at two points.

Provides a test case for generalized nonnegative scalar curvature definitions.

Abstract

In 2014, Gromov conjectured that sequences of manifolds with nonnegative scalar curvature should have subsequences which converge in some geometric sense to limit spaces with some notion of generalized nonnegative scalar curvature. In recent joint work with Changliang Wang, the authors found a sequence of warped product Riemannian metrics on $\Sph^2\times \Sph^1$ with nonnegative scalar curvature whose metric tensors converge in the $W^{1,p}$ sense for $p<2$ to an extreme warped product limit space where the warping function hits infinity at two points. Here we study this extreme limit space as a metric space and as an integral current space and prove the sequence converges in the volume preserving intrinsic flat and measured Gromov-Hausdorff sense to this space. This limit space may now be used to test any proposed definitions for generalized nonnegative scalar curvature. One does not need expertise in Geometric Measure Theory or in Intrinsic Flat Convergence to read this paper.