Geometric Convergence to an Extreme Limit Space with nonnegative scalar curvature

TL;DR

The paper addresses the problem of understanding limits of sequences of manifolds with nonnegative scalar curvature by constructing an explicit monotone warped product sequence on ${\mathbb S}^2\times{\mathbb S}^1$ whose warp function blows up at the poles. The authors prove that the sequence converges both in Gromov-Hausdorff and volume-preserving intrinsic flat senses to an extreme limit space $(\mathbb S^2\times\mathbb S^1,d_\infty)$ whose warp function is $f_\infty(r,\theta)=-2\ln\sin r+\beta$, and they compute the limit's metric completion, singular set properties, and total volume. They show $d_j$ converges monotonically to $d_\infty$, establish a homeomorphism with the isometric product, and prove that the limit is compact and not an Alexandrov, CD, or RCD space, thereby providing a concrete testbed for generalized nonnegative scalar curvature notions. The results yield both GH and VF convergence to the same limit, yield a precise description of the singular set (Hausdorff dimension 1 with infinite $\mathcal H^1$ measure and infinite unrectifiability on the fibers), and compute the limit volume $\mathcal H^3_{d_\infty}= (2\pi)^2(2\beta+4-2\ln 4)$. Together, these findings offer a rigorous framework to test and compare proposed curvature notions in low-regularity limit spaces and motivate future investigations into curvature notions in singular settings.

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.

Figures (2)

• Figure 1: Example \ref{['ex:limit']} is an extreme limit warped product space, $({\mathbb S}^2\times {\mathbb S}^1,g_\infty)$, in which the circular fibers, $r^{-1}(0)$ and $r^{-1}(\pi)$, have been stretched infinitely. Since the lengths of any curves in these fibers are infinite, we depict these fibers as circles of discrete points. Away from these two singular fibers, $({\mathbb S}^2\times {\mathbb S}^1\setminus S,g_\infty)$ is a smooth Riemannian manifold.
• Figure 2: The pointwise limit space, $({\mathbb S}^2\times {\mathbb S}^1, d_\infty)$, is a metric space in which the distances between points, even those in the circular fibers, $r^{-1}(0)$ and $r^{-1}(\pi)$, can be estimated using Lemma \ref{['LemmaDinfty-cLE']}.

Theorems & Definitions (81)

• Example 1.1
• Example 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Proposition 2.4: Volumes
• Definition 2.5: Length of a Curve
• Lemma 2.6
• Definition 2.7: Distance Function