Department of Mathematics

Rutgers, The State University of New Jersey

Hill Center - Busch Campus

110 Frelinghuysen Road

Piscataway, NJ 08854-8019, USA

Office: Hill Center 336

Email: lzlin@math.rutgers.edu

- Advisor: Professor William P. Minicozzi II
- Dissertation: On the existence of closed geodesics and uniqueness of weakly harmonic maps

- Spring 2014: 640:421 - Advanced Calculus for Engineering (Webpage for Sections 09)

- Fall 2013: 640:421 - Advanced Calculus for Engineering

- Spring 2013: 640:421 - Advanced Calculus for Engineering

- Fall 2012: 640:244 - Differential Equations for Engineering and Physics

- Summer 2012: 640:251 - Multivariable Calculus

- Spring 2012: 640:251 - Multivariable Calculus

- Fall 2011: 640:152 - Calculus II and 640:251 - Multivariable Calculus

- Summer 2010: 110.202 -
*Online*Calculus III (Instructor)- Used WebCT (Blackboard Learning System) and Elluminate Live! web conferencing software to run a completely online course

- Summer 2009: 110.202 -Calculus III (Instructor)

- Summer 2008: 110.202 -Calculus III (Instructor)

- Fall 2006 - Spring 2010: Calculus I, II and III, Differential equations and Analysis (Recitation Instructor)

**[9]** Dynamical stability of the volume preserving mean curvature flow in hyperbolic space (joint with Zheng Huang, in preparation)

**[8]** Blow-up of the mean curvature at the first singular time of the mean curvature flow (joint with Natasa Sesum, preprint, submitted); LANL(arXiv.org) link.

*Abstract: *It is conjectured that the mean curvature blows up at the first singular time of the mean curvature flow in Euclidean space, at least in dimensions less or equal to 7. We show that the mean curvature blows up at the singularities of the mean curvature flow starting from an immersed closed hypersurface with small L^2-norm of the traceless second fundamental form (observe that the initial hypersurface is not necessarily convex). As a consequence of the proof of this result we also obtain the dynamic stability of a sphere along the mean curvature flow with respect to the L^2-norm.

**[7]** Stability of the surface area preserving mean curvature flow in Euclidean space (joint with Zheng Huang, preprint, submitted); LANL(arXiv.org) link.

*Abstract: *We show that the surface area preserving mean curvature flow in Euclidean space exists for all time and converges exponentially to a round sphere, if initially the L^2-norm of the traceless second fundamental form is small (but the initial hypersurface is not necessarily convex).

**[6] **Uniformity of harmonic map heat flow at infinite time (**Analysis & PDE**, Vol. 6 (2013), No. 8, 1899--1921; APDE Link); LANL(arXiv.org) link.

*Abstract: *We show an energy convexity along the harmonic map heat flow with small initial energy and fixed boundary data on the unit 2-disk. In particular, this gives an affirmative answer to an open question asking whether such harmonic map heat flow converges uniformly in time and strongly in the W^{1,2}-topology to the unique limiting harmonic map as time goes to infinity.

**[5]** Estimates for the energy density of critical points of a class of conformally invariant variational problems (joint with Tobias Lamm, **Adv. Calc. Var.**, Volume 6, Issue 4 (Jul 2012), 391--413; ACV Link); LANL(arXiv.org) link.

Abstract: We show that the energy density of critical points of a class of conformally invariant variational problems with small energy on the unit 2-disk B_1 lies in the local Hardy space h^1(B_1). As a corollary we obtain a new proof of the energy convexity and uniqueness result for weakly harmonic maps with small energy on B_1.

**[4]** Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space (joint with Ling Xiao, **Comm. Anal. Geom.**, Volume 20, Number 5, 1061-1096, 2012; CAG Link); LANL(arXiv.org) link.

*Abstract:* We define a new version of modified mean curvature flow (MMCF) in hyperbolic space H^{n+1}, which interestingly turns out to be the natural negative L^2-gradient flow of the energy functional defined by De Silva and Spruck. We show the existence, uniqueness and convergence of the MMCF of complete embedded star-shaped hypersurfaces with fixed prescribed asymptotic boundary at infinity. As an application, we recover the existence and uniqueness of smooth complete hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, which was first shown by Guan and Spruck.

**[3]** Closed geodesics in Alexandrov spaces of curvature bounded from above (**J. Geom. Anal.**, Volume 21, Issue 2 (2011), 429-454; JGA Link); LANL(arXiv.org) link.

*Abstract:* We show a local energy convexity of W^{1,2} maps into CAT(K) spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting.

**[2]** Existence of good sweepouts on closed manifolds (joint with Lu Wang, **Proc. Amer. Math. Soc.** 138 (2010), No. 11, 4081-4088; PROC AMS Link); LANL(arXiv.org) link.

*Abstract:* We use the harmonic map heat flow to tighten the sweepout of closed curves on a closed Riemannian manifold, and we show that the tightened sweepout has the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic.

**[1]** On the existence of closed geodesics and uniqueness of weakly harmonic maps, Thesis (Ph.D.), The Johns Hopkins University (2011), 76 pp. ISBN: 978-1124-75827-5.

(1) (with Zheng Huang and Marcello Lucia, **CUNY-College of Staten Island**)
Co-Organizer of Special Session on Recent Progress in Geometric and Complex Analysis, Joint Meetings of the AMS, Baltimore, January 2014

(2) (with Xiaodong Cao and Peng Wu, **Cornell University**)
Co-Organizer of Special Session on Parabolic Evolution Equations of Geometric Type, AMS Fall Eastern Sectional Meeting, Temple University, October 2013

- Nonlinear Analysis Seminar @ Rutgers
- Differential Geometry Seminar @ CUNY Graduate Center
- Nonlinear Analysis and PDEs @ CUNY Graduate Center
- Differential Geometry & Geometric Analysis Seminar @ Princeton
- Geometry & Analysis Seminar @ Columbia University