# Jiayin Pan

I am a visiting assistant professor at the Department of Mathematics, University of California-Santa Barbara, since 2018 Fall.

I received my Ph.D. degree in 2018 from Rutgers Univerisity-New Brunswick, advised by Xiaochun Rong.

Office: South Hall 6723

My CV

A Fox

## Research Interests

Riemannian geometry, Ricci curvature and topology, Gromov-Hausdorff convergence

Research statement

## Research Articles

My preprints on arXiv

Nonnegative Ricci curvature and escape rate gap
In preparation.

On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature
arXiv:2003.01326, to appear in Geom. & Topol.

Abstract: 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 $\pi_1(M,x)$ are contained in a bounded ball, then $\pi_1(M,x)$ is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of $\pi_1(M,x)$ escape from any bounded metric balls at a sublinear rate with respect to their lengths, then $\pi_1(M,x)$ is virtually abelian.

Semi-local simple connectedness of noncollapsing Ricci limit spaces (with Guofang Wei)
arXiv:1904.06877, to appear in J. Eur. Math. Soc.

Abstract: 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.

Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups
arXiv:1809.10220

Abstract: 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 $\pi_1(M)$ 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 in terms of $n$ and $L$. 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.

Ricci curvature and isometric actions with scaling nonvanishing property (with Xiaochun Rong)
arXiv:1808.02329

Abstract: In the study of 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 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 $\delta>0$, then the maximal displacement of $f$ on the rescaled unit ball $r^{-1}B_r(p)$ is at least $\Phi(\delta,n,v)>0$ for all $r\in(0,1)$. We call this scaling $\Phi$-nonvanishing property at $p$. We study the equivariant Gromov-Hausdorff convergence of a sequence of Riemannian universal covers with abelian $\pi_1(M_i,p_i)$-actions $(\widetilde{M}_i,\tilde{p}_i,\pi_1(M_i,p_i))\rightarrow(\widetilde{X},\tilde{x},G)$, where $\pi_1(M_i,p_i)$-action is scaling $\Phi$-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 $\pi_1(M,p)$-action on $R^{-1}\widetilde{M}$ is scaling $\Phi$-nonvanishing at $\tilde{p}$ for all $R$ large, then $\pi_1(M)$ is finitely generated.

Nonnegative Ricci curvature, stability at infinity, and finite generation of fundamental groups
Geom. & Topol. 23-6 (2019) 3203–3231. DOI 10.2140/gt.2019.23.3203

Abstract: We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature. We show that if there is an integer $k$ such that any tangent cone at infinity of the Riemannian universal cover of $M$ is a metric cone, whose maximal Euclidean factor has dimension $k$, then $\pi_1(M)$ is finitely generated. In particular, this confirms the Milnor conjecture for a manifold whose universal cover has Euclidean volume growth and the unique tangent cone at infinity.

A Proof of Milnor conjecture in dimension 3
J. Reine Angew. Math. (Crelles Journal) 758 (2020) 253–260. DOI 10.1515/crelle-2017-0057

Abstract: We present a proof of Milnor conjecture in dimension $3$ based on Cheeger-Colding theory on limit spaces of manifolds with Ricci curvature bounded below. It is different from Liu’s proof that relies on minimal surface theory.

## Survey Articles

The fundamental groups of open manifolds with nonnegative Ricci curvature
arXiv:2006.00745
to appear in Special Issue of SIGMA on Scalar and Ricci Curvature

Universal covers of Ricci limit and RCD spaces (with Guofang Wei)
to appear in Differential Geometry in the Large, Cambridge University Press, 2020.

## Short notes for fun

Nonnegative Ricci curvature and virtually abelian structure
file

This short note is about fundamental groups of closed manifolds of zero sectional curvature or non-negative Ricci curvature. It includes Buser’s proof on classical Bieberbach’s theorem and Cheeger-Gromoll’s proof on virtually abelian structure. It also offers a viewpoint of virtual abelianness from virtual nilpotency.