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.

Email address: j_pan@math.ucsb.edu or jypan10@gmail.com
Office: South Hall 6723


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)(M,x) of nonnegative Ricci curvature, if all the minimal geodesic loops at xx that represent elements of π1(M,x)\pi_1(M,x) are contained in a bounded ball, then π1(M,x)\pi_1(M,x) is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of π1(M,x)\pi_1(M,x) escape from any bounded metric balls at a sublinear rate with respect to their lengths, then π1(M,x)\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 XX be a non-collapsing Ricci limit space and let xXx\in X. We show that for any ϵ>0\epsilon>0, there is r>0r>0 such that every loop in Bt(x)B_t(x) is contractible in B(1+ϵ)t(x)B_{(1+\epsilon)t}(x), where t(0,r]t\in(0,r]. In particular, XX is semi-locally simply connected.

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

Abstract: We study the fundamental group of an open nn-manifold MM of nonnegative Ricci curvature with additional stability condition on M~\widetilde{M}, the Riemannian universal cover of MM. We prove that if any tangent cone of M~\widetilde{M} at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric space, then π1(M)\pi_1(M) is finitely generated and contains a normal abelian subgroup of finite index; if in addition M~\widetilde{M} has Euclidean volume growth of constant at least LL, then we can bound the index of that abelian subgroup in terms of nn and LL. Our result implies that if M~\widetilde{M} has Euclidean volume growth of constant at least 1ϵ(n)1−\epsilon(n), then π1(M)\pi_1(M) is finitely generated and C(n)C(n)-abelian.

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

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, vol(B1(p))v>0\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 ff on B1(p)B_1(p) is at least δ>0\delta>0, then the maximal displacement of ff on the rescaled unit ball r1Br(p)r^{-1}B_r(p) is at least Φ(δ,n,v)>0\Phi(\delta,n,v)>0 for all r(0,1)r\in(0,1). We call this scaling Φ\Phi-nonvanishing property at pp. We study the equivariant Gromov-Hausdorff convergence of a sequence of Riemannian universal covers with abelian π1(Mi,pi)\pi_1(M_i,p_i)-actions (M~i,p~i,π1(Mi,pi))(X~,x~,G)(\widetilde{M}_i,\tilde{p}_i,\pi_1(M_i,p_i))\rightarrow(\widetilde{X},\tilde{x},G), where π1(Mi,pi)\pi_1(M_i,p_i)-action is scaling Φ\Phi-nonvanishing at pi~\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 MM of non-negative Ricci curvature, if the universal cover M~\widetilde{M} has Euclidean volume growth and π1(M,p)\pi_1(M,p)-action on R1M~R^{-1}\widetilde{M} is scaling Φ\Phi-nonvanishing at p~\tilde{p} for all RR large, then π1(M)\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 nn-manifold MM of nonnegative Ricci curvature. We show that if there is an integer kk such that any tangent cone at infinity of the Riemannian universal cover of MM is a metric cone, whose maximal Euclidean factor has dimension kk, then π1(M)\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 33 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
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

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.