Jiayin Pan

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

Email address: j_pan@math.ucsb.edu or jypan10@gmail.com
Office: South Hall 6723
Office hour 2019 Spring: Monday 1-4 pm

My CV

A Fox

Research

Research interests: Riemannian geometry, Ricci curvature and topology, Gromov-Hausdorff convergence
My preprints on arXiv

Publications and Preprints

Semi-local simple connectedness of noncollapsing Ricci limit spaces. (Joint work with Guofang Wei) arXiv:1904.06877

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. arXiv:1809.10220

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. (Joint work 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, 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. arXiv:1710.05498, to appear in Geom. Topol.

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., 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.

Short notes for fun

Nonnegative Ricci curvature and virtually abelian structure. file

This short note includes Buser’s proof on classical Bieberbach’s theorem and Cheeger-Gromoll’s proof on virtually abelian structure. It also offers a viewpoint of virtually abelian structure from virtually nilpotent structure.