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.

**Email address:** j_pan@math.ucsb.edu or jypan10@gmail.com

**Office:** South Hall 6723

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

*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.

*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.

*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.