Ricci Limit Research Articles

Ricci limit flows and weak solutions

In this paper we reconcile several different approaches to Ricci flow through singularities that have been proposed over the last few years by Kleiner–Lott, Haslhofer–Naber and Bamler. Specifically, we prove that every noncollapsed limit of Ricci flows, as provided by Bamler’s precompactness theorem, as well as every singular Ricci flow of Kleiner–Lott, is a weak solution in the sense of Haslhofer–Naber. We also generalize all path-space estimates of Haslhofer–Naber to the setting of noncollapsed Ricci limit flows. The key step to establish these results is a new hitting estimate for Brownian motion. A fundamental difficulty, in stark contrast to all prior hitting estimates in the literature, is the lack of lower heat kernel bounds under Ricci flow. To overcome this, we introduce a novel approach to hitting estimates that compensates for the lack of lower heat kernel bounds by making use of the heat kernel geometry of space-time.

Ricci limit spaces are semi-locally simply connected

Ricci limit spaces are semi-locally simply connected

Optimal diameter estimate of three-dimensional Ricci limit spaces

In this note, we prove that positive scalar curvature can pass to three dimensional Ricci limit spaces of non-negative Ricci curvature when it splits off a line. As a corollary, we obtain an optimal Bonnet-Myers type upper bound. Moreover, we obtain a similar statement in all dimensions for Alexandrov spaces of non-negative curvature.

RCD*(K,N) spaces are semi-locally simply~connected

Abstract It was shown in [A. Mondino and G. Wei, On the universal cover and the fundamental group of an RCD * ⁢ ( K , N ) {\rm RCD}^{*}(K,N) -space, J. reine angew. Math. 753 2019, 211–237] that any RCD * ⁢ ( K , N ) {\mathrm{RCD}^{*}(K,N)} space ( X , d , 𝔪 ) {(X,d,\mathfrak{m})} has a universal cover. We prove that for any point x ∈ X {x\in X} and R > 0 {R>0} , there exists r < R {r<R} such that any loop in B r ⁢ ( x ) {B_{r}(x)} is contractible in B R ⁢ ( x ) {B_{R}(x)} ; in particular, X is semi-locally simply connected and the universal cover of X is simply connected. This generalizes the earlier work in [J. Wang, Ricci limit spaces are semi-locally simply connected, preprint 2021] that any Ricci limit space is semi-locally simply connected.

Singular Weyl’s law with Ricci curvature bounded below

We establish two surprising types of Weyl’s laws for some compact RCD ⁡ ( K , N ) \operatorname {RCD}(K, N) /Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for RCD ⁡ ( K , N ) \operatorname {RCD}(K,N) spaces. Our results depend crucially on analyzing and developing important properties of the examples constructed in Pan and Wei [Geom. Funct. Anal. 32 (2022), pp. 676–685], showing them isometric to the α \alpha -Grushin halfplanes. Of independent interest, this also allows us to provide counterexamples to conjectures in Cheeger and Colding [J. Differential Geom. 46 (1997), pp. 406–480] and Kapovitch, Kell, and Ketterer [Math. Z. 301 (2022), pp. 3469–3502].

Removable sets and [formula omitted]-uniqueness on manifolds and metric measure spaces

We study symmetric diffusion operators on metric measure spaces. Our main question is whether essential self-adjointness or Lp-uniqueness are preserved under the removal of a small closed set from the space. We provide characterizations of the critical size of removed sets in terms of capacities and Hausdorff dimension without any further assumption on removed sets. As a key tool we prove a non-linear truncation result for potentials of nonnegative functions. Our results are robust enough to be applied to Laplace operators on general Riemannian manifolds as well as sub-Riemannian manifolds and metric measure spaces satisfying curvature-dimension conditions. For non-collapsing Ricci limit spaces with two-sided Ricci curvature bounds we observe that the self-adjoint Laplacian is already fully determined by the classical Laplacian on the regular part.

The Grushin hemisphere as a Ricci limit space with curvature ≥1

The Grushin sphere is an almost-Riemannian manifold that degenerates along its equator. We construct a sequence of Riemannian metrics on a sphere S m + n S^{m+n} with R i c ≥ 1 Ric\ge 1 such that its Gromov-Hausdorff limit is the n n -dimensional Grushin hemisphere.

The rigidity of sharp spectral gap in non-negatively curved spaces

We extend the celebrated rigidity of the sharp first spectral gap underRic≥0 to compact infinitesimally Hilbertian spaces with non-negative (weak, also called synthetic) Ricci curvature and bounded (synthetic) dimension i.e. to so-called compact RCD0,N spaces; this is a category of metric measure spaces which in particular includes (Ricci) non-negatively curved Riemannian manifolds, Alexandrov spaces, Ricci limit spaces, Bakry–Émery manifolds along with products, certain quotients and measured Gromov–Hausdorff limits of such spaces. In precise terms, we show in such spaces,λ1=π2diam2if and only if the space is one dimensional with a constant density function. We use new techniques mixing Sobolev theory and singular 1D-localization which might also be of independent interest. As a consequence of the rigidity in the singular setting, we also derive almost rigidity results.

Local control on the geometry in 3D Ricci flow

The geometry of a ball within a Riemannian manifold is coarsely controlled if it has a lower bound on its Ricci curvature and a positive lower bound on its volume. We prove that such coarse local geometric control must persist for a definite amount of time under three-dimensional Ricci flow, and leads to local $C/t$ decay of the full curvature tensor, irrespective of what is happening beyond the local region. As a by-product, our results generalise the Pseudolocality theorem of Perelman [19, §10.1 and §10.5] and Tian-Wang [25] in this dimension by not requiring the Ricci curvature to be almostpositive, and not asking the volume growth to be almost-Euclidean. Our results also have applications to the topics of starting Ricci flow with manifolds of unbounded curvature, to the use of Ricci flow as a mollifier, and to the well-posedness of Ricci flow starting with Ricci limit spaces. In [24] we use results from this paper to prove that 3D Ricci limit spaces are locally bi-Hölder equivalent to smooth manifolds, going beyond a full resolution of the conjecture of Anderson, Cheeger, Colding and Tian in this dimension.

A Lorentzian analog for Hausdorff dimension and measure

We define a one-parameter family of canonical volume measures on Lorentzian (pre-)length spaces. In the Lorentzian setting, this allows us to define a geometric dimension - akin to the Hausdorff dimension for metric spaces - that distinguishes between e.g. spacelike and null subspaces of Minkowski spacetime. The volume measure corresponding to its geometric dimension gives a natural reference measure on a synthetic or limiting spacetime, and allows us to define what it means for such a spacetime to be collapsed (in analogy with metric measure geometry and the theory of Riemannian Ricci limit spaces). As a crucial tool we introduce a doubling condition for causal diamonds and a notion of causal doubling measures. Moreover, applications to continuous spacetimes and connections to synthetic timelike curvature bounds are given.

Some topological results of Ricci limit spaces

We study the topology of a Ricci limit space ( X , p ) (X,p) , which is the Gromov-Hausdorff limit of a sequence of complete n n -manifolds ( M i , p i ) (M_i, p_i) with R i c ≥ − ( n − 1 ) \mathrm {Ric}\ge -(n-1) . Our first result shows that, if M i M_i has Ricci bounded covering geometry, i.e. the local Riemannian universal cover is non-collapsed, then X X is semi-locally simply connected. In the process, we establish a slice theorem for isometric pseudo-group actions on a closed ball in the Ricci limit space. In the second result, we give a description of the universal cover of X X if M i M_i has a uniform diameter bound; this improves a result by Ennis and Wei [Differential Geom. Appl. 24 (2006), pp. 554-562].

Minimal hypersurfaces in manifolds of Ricci curvature bounded below

Abstract In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding’s method in [T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 1997, 3, 477–501]. With Cheeger–Colding theory, we obtain the Laplacian comparison for limits of distance functions from minimal hypersurfaces in the version of Ricci limit space. As an application, if a sequence of minimal hypersurfaces converges to a metric cone C ⁢ Y × ℝ n - k {CY\times\mathbb{R}^{n-k}} ( 2 ≤ k ≤ n {2\leq k\leq n} ) in a non-collapsing metric cone C ⁢ X × ℝ n - k {CX\times\mathbb{R}^{n-k}} obtained from ambient manifolds of almost nonnegative Ricci curvature, then we can prove a Frankel property for the cross section Y of CY. Namely, Y has only one connected component in X.

Global regularity of three-dimensional Ricci limit spaces

Miles Simon and the second author, in their recent work [Geom. Topol. 25 (2021), pp. 913–948], established a local bi-Hölder correspondence between weakly noncollapsed Ricci limit spaces in three dimensions and smooth manifolds. In particular, any open ball of finite radius in such a limit space must be bi-Hölder homeomorphic to some open subset of a complete smooth Riemannian three-manifold. In this work we build on the technology from Simon and the second author in [J. Differential Geometry, to appear] and [Geom. Topol. 25, (2021), pp. 913–948] to improve this local correspondence to a global-local correspondence. That is, we construct a smooth three-manifold M , M , and prove that the entire (weakly) noncollapsed three-dimensional Ricci limit space is homeomorphic to M M via a globally-defined homeomorphism that is bi-Hölder once restricted to any compact subset. Here the bi-Hölder regularity is with respect to the distance d g d_g on M , M, where g g is any smooth complete metric on M M . A key step in our proof is the construction of local pyramid Ricci flows, existing on uniform regions of spacetime, that are inspired by Hochard’s partial Ricci flows in the paper by Raphaël Hochard [Short-time existence of the Ricci flow on complete, non-collapsed 3-manifolds with Ricci curvature bounded from below, https://arxiv.org/abs/1603.08726, 2016]. Suppose ( M , g 0 , x 0 ) \left ( M , g_0 , x_0 \right ) is a complete smooth pointed Riemannian three-manifold that is (weakly) noncollapsed and satisfies a lower Ricci bound. Then, given any k ∈ N , k \in \mathbb {N}, we construct a smooth Ricci flow g ( t ) g(t) living on a subset of spacetime that contains, for each j ∈ { 1 , … , k } j \in \left \{1 , \ldots , k \right \} , a cylinder B g 0 ( x 0 , j ) × [ 0 , T j ] \mathbb {B}_{g_0} \left ( x_0 , j \right )\times [0,T_j] , where T j T_j is dependent only on the Ricci lower bound, the (weakly) noncollapsed volume lower bound and the radius j j (in particular independent of k k ) and with the property that g ( 0 ) = g 0 g(0)=g_0 throughout B g 0 ( x 0 , k ) \mathbb {B}_{g_0} \left ( x_0 , k \right ) .

Examples of Ricci limit spaces with non-integer Hausdorff dimension

We give the first examples of collapsing Ricci limit spaces on which the Hausdorff dimension of the singular set exceeds that of the regular set; moreover, the Hausdorff dimension of these spaces can be non-integers. This answers a question of Cheeger-Colding about collapsing Ricci limit spaces.

Semi-local simple connectedness of non-collapsing Ricci limit spaces

Let $X$ be a non-collapsing Ricci limit space and let $x\in X$. We show that for any $\epsilon>0$, there is $r>0$ such that every loop in $B_t(x)$ is contractible in $B_{(1+\epsilon)t}(x)$, where $t\in(0,r]$. In particular, $X$ is semi-locally simply connected.

Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces

We use Ricci flow to obtain a local bi-Holder correspondence between Ricci limit spaces in three dimensions and smooth manifolds. This is more than a complete resolution of the three-dimensional case of the conjecture of Anderson-Cheeger-Colding-Tian, describing how Ricci limit spaces in three dimensions must be homeomorphic to manifolds, and we obtain this in the most general, locally non-collapsed case. The proofs build on results and ideas from recent papers of Hochard and the current authors.

Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K, N) metric measure spaces; regularity is understood with respect to a newly defined quasi‐metric built from the Green function of the Laplacian. Its main application is that RCD(K, N) spaces have constant dimension. In this way we generalize to such an abstract framework a result proved by Colding‐Naber for Ricci limit spaces, introducing ingredients that are new even in the smooth setting. © 2019 Wiley Periodicals, Inc.

Elliptic PDEs on Compact Ricci Limit Spaces and Applications

In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrodinger operators, generalized Yamabe constants and eigenvalues of the Hodge Laplacian, with respect to the Gromov-Hausdorff topology. We apply these to the study of second-order differential calculus on such limit spaces.

Angles between Curves in Metric Measure Spaces

AbstractThe goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on RCD*(K, N) metric measure spaces. As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian manifolds, Ricci limit spaces and Alexandrov spaces.

On the universal cover and the fundamental group of an RCD*(K,N)-space

Abstract The main goal of the paper is to prove the existence of the universal cover for 𝖱𝖢𝖣 * ⁢ ( K , N ) {\mathsf{RCD}^{*}(K,N)} -spaces. This generalizes earlier work of [43, 44] on the existence of universal covers for Ricci limit spaces. As a result, we also obtain several structure results on the (revised) fundamental group of 𝖱𝖢𝖣 * ⁢ ( K , N ) {\mathsf{RCD}^{*}(K,N)} -spaces. These are the first topological results for 𝖱𝖢𝖣 * ⁢ ( K , N ) {\mathsf{RCD}^{*}(K,N)} -spaces without extra structural-topological assumptions (such as semi-local simple connectedness).