Optimal Transport: A Semi-Discrete Approach

Herein we present open problems and survey examples and theorems concerning sequences of Riemannian manifolds with uniform lower bounds on scalar curvature and their limit spaces. Examples of Gromov and of Ilmanen which naturally ought to have certain limit spaces do not converge with respect to smooth or Gromov-Hausdorff convergence. Thus we focus here on the notion of Intrinsic Flat convergence, developed jointly with Wenger. This notion has been applied successfully to study sequences that arise in General Relativity. Gromov has suggested it should be applied in other settings as well. We first review intrinsic flat convergence, its properties, and its compactness theorems, before presenting the applications and the open problems.

Scalar Curvature and Intrinsic Flat Convergence

One of the most powerful theorems in metric geometry is the Arzela-Ascoli Theorem which provides a continuous limit for sequences of equicontinuous functions between two compact spaces. This theorem has been extended by Gromov and Grove-Petersen to sequences of functions with varying domains and ranges where the domains and the ranges respectively converge in the Gromov-Hausdorff sense to compact limit spaces. However such a powerful theorem does not hold when the domains and ranges only converge in the intrinsic flat sense due to the possible disappearance of points in the limit. 
In this paper two Arzela-Ascoli Theorems are proven for intrinsic flat converging sequences of manifolds: one for uniformly Lipschitz functions with fixed range whose domains are converging in the intrinsic flat sense, and one for sequences of uniformly local isometries between spaces which are converging in the intrinsic flat sense. A basic Bolzano-Weierstrass Theorem is proven for sequences of points in such sequences of spaces. In addition it is proven that when a sequence of manifolds has a precompact intrinsic flat limit then the metric completion of the limit is the Gromov-Hausdorff limit of regions within those manifolds. Applications and suggested applications of these results are described in the final section of this paper.

Intrinsic Flat Arzela-Ascoli Theorems

We consider smooth Riemannian manifolds with nonnegative Ricci curvature and smooth boundary. First we prove a global Laplacian comparison theorem in the barrier sense for the distance to the boundary. We apply this theorem to obtain volume estimates of the manifold and of regions of the manifold near the boundary depending upon an upper bound on the area and on the inward pointing mean curvature of the boundary. We prove that families of oriented manifolds with uniform bounds of this type are compact with respect to the Sormani–Wenger Intrinsic Flat (SWIF) distance.

Volumes and limits of manifolds with Ricci curvature and mean curvature bounds

We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric neighborhoods of the boundaries. We conclude several rigidity theorems. As one of them, we obtain a volume growth rigidity theorem. We also show a splitting theorem of Cheeger-Gromoll type under the assumption of the existence of a single ray.

/pdf/rigidity-of-manifolds-with-boundary-under-a-lower-ricci-1j40iywssf.pdf

Rigidity of manifolds with boundary under a lower Ricci curvature bound

By works of Schoen–Yau and Gromov–Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to 0. We prove flat and intrinsic flat subconvergence to a flat torus for noncollapsing sequences of 3-dimensional tori 
$$M_j$$

 that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound and scalar curvature bounds of the form 
$$R_{g_{M_j}} \ge -1/j$$

. We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang–Lee, Huang–Lee–Sormani and Allen–Perales–Sormani. Furthermore, our results also hold when the condition on the scalar curvature of a torus 
$$(M, g_M)$$

 is replaced by a bound on the quantity 
$$-\int _T \min \{R_{g_M},0\} d{\mathrm {vol}_{g_T}}$$

, where 
$$M=\text {graph}(f)$$

, 
$$f: T \rightarrow \mathbb {R}$$

 and 
$$(T,g_T)$$

 is a flat torus. Using arguments developed by Alaee, McCormick and the first named author after this work was completed, our results hold for dimensions 
$$n \ge 4$$

 as well.

Stability of graphical tori with almost nonnegative scalar curvature

This survey reviews precompactness theorems for classes of Riemannian manifolds with boundary. We begin with the works of Kodani, Anderson-Katsuda-Kurylev-Lassas-Taylor and Wong. We then present new results of Knox and the author with Sormani.

A survey on the Convergence of Manifolds with Boundary

For $n$-dimensional Riemannian manifolds $M$ with Ricci curvature bounded below by $-(n-1)$, the volume entropy is bounded above by $n-1$ If $M$ is compact, it is known that the equality holds if and only if $M$ is hyperbolic We extend this result to $\mathsf{RCD}^{\ast}(-(N-1),N)$ spaces While the upper bound is straightforward, the rigidity case is quite involved due to the lack of a smooth structure in $\mathsf{RCD}^{\ast}$ spaces As an application we obtain an almost rigidity result which partially recovers a result by Cheng-Rong-Xu for Riemannian manifolds

Maximal volume entropy rigidity for $\mathsf{RCD}^*(-(N-1),N)$ spaces

By works of Schoen-Yau and Gromov-Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to $0$. We prove flat and intrinsic flat subconvergence to a flat torus for noncollapsing sequences of $3$-dimensional tori $M_j$ that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound and scalar curvature bounds of the form $R_{g_{M_j}} \geq -1/j$. We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang-Lee, Huang-Lee-Sormani and Allen-Perales-Sormani. Furthermore, our results also hold when the condition on the scalar curvature of a torus $(M, g_M)$ is replaced by a bound on the quantity $-\int_T \min\{R_{g_M},0\} d{\mbox{vol}_{g_T}}$, where $M=\mbox{graph}(f)$, $f: T \to \mathbb R$ and $(T,g_T)$ is a flat torus. Using arguments developed by Alaee, McCormick and the first named author after this work was completed, our results hold for dimensions $n \geq 4$ as well.

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Papers

Volumes and limits of manifolds with Ricci curvature and mean curvature bounds

Stability of graphical tori with almost nonnegative scalar curvature

A survey on the Convergence of Manifolds with Boundary

Maximal volume entropy rigidity for $\mathsf{RCD}^*(-(N-1),N)$ spaces

Stability of graphical tori with almost nonnegative scalar curvature