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Creators/Authors contains: "Pan, Jiayin"

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  1. (, Advances in Mathematics)
    We study the fundamental group of an open $$n$$-manifold $$M$$ of nonnegative Ricci curvature with additional stability conditions on $$\widetilde{M}$$, the Riemannian universal cover of $$M$$. We prove that if every asymptotic cone of $$\widetilde{M}$$ is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric cone, then $$\pi_1(M)$$ 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 by a constant $C(n,L)$. In particular, our result implies that if $$\widetilde{M}$$ has Euclidean volume growth of constant at least $$1-\epsilon(n)$$, then $$\pi_1(M)$$ is finitely generated and $C(n)$-abelian. 
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    Free, publicly-accessible full text available July 1, 2026
  2. (, Journal of Mathematical Study)
    For a Gromov-Hausdorff convergent sequence of closed manifolds $$M_i^n\ghto X$$ with $$\Ric\ge-(n-1)$$, $$ \mathrm{diam}(M_i)\le D$$, and $$\mathrm{vol}(M_i)\ge v>0,$$ we study the relation between $$\pi_1(M_i)$$ and $$X$$. It was known before that there is a surjective homomorphism $$\phi_i:\pi_1(M_i)\to \pi_1(X)$$ by the work of Pan--Wei. In this paper, we construct a surjective homomorphism $$\psi_i$$ from the interior of the effective regular set in $$X$$ back to $$M_i$$. These surjective homomorphisms $$\phi_i$$ and $$\psi_i$$ are natural in the sense that their composition $$\phi_i \circ \psi_i$$ is exactly the homomorphism induced by the inclusion map from the effective regular set to $$X$$. 
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    Free, publicly-accessible full text available January 1, 2026
  3. ; (, Advances in Mathematics)
    We study the rigidity problems for open (complete and noncompact) $$n$$-manifolds with nonnegative Ricci curvature. We prove that if an asymptotic cone of $$M$$ properly contains a Euclidean $$\mathbb{R}^{k-1}$$, then the first Betti number of $$M$$ is at most $n-k$; moreover, if equality holds, then $$M$$ is flat. Next, we study the geometry of the orbit $$\Gamma\tilde{p}$$, where $$\Gamma=\pi_1(M,p)$$ acts on the universal cover $$(\widetilde{M},\tilde{p})$$. Under a similar asymptotic condition, we prove a geometric rigidity in terms of the growth order of $$\Gamma\tilde{p}$$. We also give the first example of a manifold $$M$$ of $$\mathrm{Ric}>0$$ and $$\pi_1(M)=\mathbb{Z}$$ but with a varying orbit growth order. 
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    Free, publicly-accessible full text available December 1, 2025
  4. ; ; ; (, Transactions of the American Mathematical Society, Series B)
    We establish two surprising types of Weyl’s laws for some compact RCD ⁡<#comment/> ( 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 ⁡<#comment/> ( 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<#comment/> \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]. 
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