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Jiayin Pan

Publications and source records attributed to Jiayin Pan.

At least 19 recordsLinked to original sources
arXiv2026

Nonnegative Ricci curvature and virtual abelianness in dimensions less than 12

For any complete Riemannian manifold $M^n$ with nonnegative Ricci curvature and sublinear diameter growth, we establish a dimensional constraint $n\ge 4s(s-1)+k+1$ if the fundamental group $\pi_1(M)$ contains a torsion-free nilpotent subgroup of rank $k$ and step $s\ge 2$. As a consequence, if such a manifold $M$ has dimension $n<12$, then $\pi_1(M)$ is almost abelian. The proof is based on a dimensional estimate for $\mathrm{RCD}(0,N)$ spaces admitting $\mathbb{R}$-orbits of large Hausdorff dimension.

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arXiv2026

Complete manifolds with nonnegative Ricci curvature and slow relative volume growth

For any complete and noncompact manifold $M$ with $\mathrm{Ric}\ge 0$, we define a function $\mathrm{RV}(s)$ that describes the growth of relative volume asymptotically $$\mathrm{RV}(s)=\limsup_{r\to\infty} \dfrac{\mathrm{vol} B_{rs}(p)}{\mathrm{vol} B_r(p)},\quad s\ge 1.$$ Then we study the fundamental groups of such manifolds with slow relative volume growth and sublinear diameter growth. We show that if $\mathrm{RV}(s)\ll s^2$ as $s\to\infty$, then $\pi_1(M)$ is almost abelian; if $\mathrm{RV}(s)\ll s^{1+\delta}$ for some $\delta\in (0,1)$ and the Ricci curvature is positive at a point, then $\pi_1(M)$ is finite. These results generalize our previous work on complete manifolds with $\mathrm{Ric}\ge 0$ and linear (minimal) volume growth.

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arXiv2025

Universal non-CD of sub-Riemannian manifolds

We prove that a sub-Riemannian manifold equipped with a full-support Radon measure is never $\mathrm{CD}(K,N)$ for any $K\in \mathbb{R}$ and $N\in (1,\infty)$ unless it is Riemannian. This generalizes previous non-CD results for sub-Riemannian manifolds, where a measure with smooth and positive density is considered. Our proof is based on the analysis of the tangent cones and the geodesics within. Secondly, we construct new $\mathrm{RCD}$ structures on $\mathbb{R}^n$, named cone-Grushin spaces, that fail to be sub-Riemannian due to the lack of a scalar product along a curve, yet exhibit characteristic features of sub-Riemannian geometry, such as horizontal directions, large Hausdorff dimension, and inhomogeneous metric dilations.

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arXiv2025

Nonnegative Ricci curvature, nilpotency, and Hausdorff dimension

Let $M$ be an open (complete and non-compact) manifold with $\mathrm{Ric}\ge 0$ and escape rate not $1/2$. It is known that under these conditions, the fundamental group $π_1(M)$ has a finitely generated torsion-free nilpotent subgroup $\mathcal{N}$ of finite index, as long as $π_1(M)$ is an infinite group. We show that the nilpotency step of $\mathcal{N}$ must be reflected in the asymptotic geometry of the universal cover $\widetilde{M}$, in terms of the Hausdorff dimension of an isometric $\mathbb{R}$-orbit: there exist an asymptotic cone $(Y,y)$ of $\widetilde{M}$ and a closed $\mathbb{R}$-subgroup $L$ of the isometry group of $Y$ such that its orbit $Ly$ has Hausdorff dimension at least the nilpotency step of $\mathcal{N}$. This resolves a question raised by Wei and the author.

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arXiv2024

On the topology of manifolds with nonnegative Ricci curvature and linear volume growth

Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and linear volume growth. First, we show that the fundamental group of such a manifold contains a subgroup $\mathbb{Z}^k$ of finite index, where $0\le k\le n-1$. Second, we prove that if the Ricci curvature is positive everywhere, then the fundamental group is finite. The proofs are based on an analysis of the equivariant asymptotic geometry of successive covering spaces and a plane/halfplane rigidity result for RCD spaces.

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arXiv2024

Ricci curvature and fundamental groups of effective regular sets

For a Gromov-Hausdorff convergent sequence of closed manifolds $M_i^n\overset{GH}\longrightarrow X$ with $\mathrm{Ric}\ge-(n-1)$, $\mathrm{diam}(M_i)\le D$, and $\mathrm{vol}(M_i)\ge v>0$, we study the relation between $π_1(M_i)$ and $X$. It was known before that there is a surjective homomorphism $ϕ_i:π_1(M_i)\to π_1(X)$ by the work of Pan-Wei. In this paper, we construct a surjective homomorphism from the interior of the effective regular set in $X$ back to $M_i$, that is, $ψ_i:π_1(\mathcal{R}_{ε,δ}^\circ)\to π_1(M_i)$. These surjective homomorphisms $ϕ_i$ and $ψ_i$ are natural in the sense that their composition $ϕ_i \circ ψ_i$ is exactly the homomorphism induced by the inclusion map $\mathcal{R}_{ε,δ}^\circ \hookrightarrow X$.

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arXiv2023

Singular Weyl's law with Ricci curvature bounded below

We establish two surprising types of Weyl's laws for some compact $\mathrm{RCD}(K, N)$/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 $\mathrm{RCD}(K,N)$ spaces. Our results depends crucially on analyzing and developing important properties of the examples constructed by the last two authors, showing them isometric to the $α$-Grushin halfplanes. Of independent interest, this also allows us to provide counterexamples to conjectures by Cheeger-Colding and by Kapovitch-Kell-Ketterer.

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arXiv2022

Nonnegative Ricci curvature, metric cones, and virtual abelianness

Let $M$ be an open $n$-manifold with nonnegative Ricci curvature. We prove that if its escape rate is not $1/2$ and its Riemannian universal cover is conic at infinity, that is, every asymptotic cone $(Y,y)$ of the universal cover is a metric cone with vertex $y$, then $π_1(M)$ contains an abelian subgroup of finite index. If in addition the universal cover has Euclidean volume growth of constant at least $L$, we can further bound the index by a constant $C(n,L)$.

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arXiv2021

Nonnegative Ricci curvature and escape rate gap

Let $M$ be an open $n$-manifold of nonnegative Ricci curvature and let $p\in M$. We show that if $(M,p)$ has escape rate less than some positive constant $ε(n)$, that is, minimal representing geodesic loops of $π_1(M,p)$ escape from any bounded balls at a small linear rate with respect to their lengths, then $π_1(M,p)$ is virtually abelian. This generalizes the author's previous work, where the zero escape rate is considered.

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arXiv2021

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.

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arXiv2021

Some topological results of Ricci limit spaces

We study the topology of a Ricci limit space $(X,p)$, which is the Gromov-Hausdorff limit of a sequence of complete $n$-manifolds $(M_i, p_i)$ with $\mathrm{Ric}\ge -(n-1)$. Our first result shows that, if $M_i$ has Ricci bounded covering geometry, i.e. the local Riemannian universal cover is non-collapsed, then $X$ 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 $X$ if $M_i$ has a uniform diameter bound.

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arXiv2020

On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature

A consequence of the Cheeger-Gromoll splitting theorem states that for any open manifold $(M,x)$ of nonnegative Ricci curvature, if all the minimal geodesic loops at $x$ that represent elements of $π_1(M,x)$ are contained in a bounded ball, then $π_1(M,x)$ is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of $π_1(M,x)$ escape from any bounded metric balls at a sublinear rate with respect to their lengths, then $π_1(M,x)$ is virtually abelian.

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arXiv2019

Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature with additional stability condition on $\widetilde{M}$, the Riemannian universal cover of $M$. We prove that if any tangent cone of $\widetilde{M}$ at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric space, then $π_1(M)$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $\widetilde{M}$ has Euclidean volume growth of constant at least $L$, then we can bound the index of that abelian subgroup in terms of $n$ and $L$. In particular, our result implies that if $\widetilde{M}$ has Euclidean volume growth of constant at least $1-ε(n)$, then $π_1(M)$ is finitely generated and $C(n)$-abelian.

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arXiv2018

Ricci curvature and isometric actions with scaling nonvanishing property

In the study manifolds of Ricci curvature bounded below, a stumbling obstruction is the lack of links between large-scale geometry and small-scale geometry at a fixed reference point. There have been few links (volume, dimension) when the unit ball at the point is not collapsed, that is, $\mathrm{vol}(B_1(p))\ge v>0$. In this paper, we conjecture a new link in terms of isometries: if the maximal displacement of an isometry $f$ on $B_1(p)$ is at least $δ>0$, then the maximal displacement of $f$ on the rescaled unit ball $r^{-1}B_r(p)$ is at least $Φ(δ,n,v)>0$ for all $r\in(0,1)$. We call this scaling $Φ$-nonvanishing property at $p$. We study the equivariant Gromov-Hausdorff convergence of a sequence of Riemannian universal covers with abelian $π_1(M_i,p_i)$-actions $(\widetilde{M}_i,\tilde{p}_i,π_1(M_i,p_i))\overset{GH}\longrightarrow(\widetilde{X},\tilde{p},G)$, where $π_1(M_i,p_i)$-action is scaling $Φ$-nonvanishing at $\tilde{p_i}$. We establish a dimension monotonicity on the limit group associated to any rescaling sequence. As one of the applications, we prove that for an open manifold $M$ of non-negative Ricci curvature, if the universal cover $\widetilde{M}$ has Euclidean volume growth and $π_1(M,p)$-action on $R^{-1}\widetilde{M}$ is scaling $Φ$-nonvanishing at $\tilde{p}$ for all $R$ large, then $π_1(M)$ is finitely generated.

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