Research & Publications

Research interests: Geometric analysis and Partial differential equations, more precisely, geometric flows including mean curvature flow and harmonic map heat flow, harmonic maps, minimal surfaces, surfaces of constant mean curvature, and min-max theory.


Preprints

[17] Modified mean curvature flow and CMC foliation conjecture in almost Fuchsian manifolds (joint with Zheng Huang and Zhou Zhang), arXiv:2311.04298

Abstract:There has been a conjecture, often attributed to Thurston, which asserts that every almost Fuchsian manifold is foliated by closed incompressible constant mean curvature (CMC) surfaces. In this paper, for a certain class of almost Fuchsian manifolds, we prove the long-time existence and convergence of the modified mean curvature flow, which was first introduced by Xiao and the second named author in \cite{LX12}. As an application, we confirm Thurston's CMC foliation conjecture for such a subclass of almost Fuchsian manifolds.

[16] On the uniqueness of conformal-harmonic maps (joint with Jingyong Zhu), arXiv:2206.05874

Abstract: Motivated by the theory of harmonic maps on Riemannian surfaces, conformal-harmonic maps between two Riemannian manifolds M and N were introduced in search of a natural notion of "harmonicity" for maps defined on a general even dimensional Riemannian manifold M. They are critical points of a conformally invariant energy functional and reassemble the GJMS operators when the target is the set of real or complex numbers. On a four dimensional manifold, conformal-harmonic maps are the conformally invariant counterparts of the intrinsic bi-harmonic maps and a mapping version of the conformally invariant Paneitz operator for functions. In this paper, we consider conformal-harmonic maps from certain locally conformally flat 4-manifolds into spheres. We prove a quantitative uniqueness result for such conformal-harmonic maps as an immediate consequence of convexity for the conformally-invariant energy functional. To this end, we are led to prove a version of second order Hardy inequality on manifolds, which may be of independent interest.


Published Journal Articles

[15] Existence of polyharmonic maps in critical dimensions (joint with Weiyong He and Ruiqi Jiang), Journal of Functional Analysis, Volume 285, Issue 5, 1 September 2023, 110020, arXiv link.

[14] Stability of the volume preserving mean curvature flow in hyperbolic space (joint with Zheng Huang and Zhou Zhang), Peking Mathematical Journal (2023); arXiv link.

[13] Energy convexity of intrinsic bi-harmonic maps and applications I: spherical target (joint with Paul Laurain), Journal für die reine und angewandte Mathematik (Crelle's Journal), vol. 2021, no. 772, 2021, pp. 53-81; arXiv link.

[12] Mean curvature flow of star-shaped hypersurfaces, Communications in Analysis and Geometry. Volume 28, Number 6, 1315–1336, 2020; arXiv link.

[11] Min-max minimal disks with free boundary in Riemannian manifolds (joint with Ao Sun and Xin Zhou), Geometry & Topology 24-1 (2020), 471--532. DOI 10.2140/gt.2020.24.471; arXiv link.

[10] Mean curvature flow in Fuchsian manifolds (joint with Zheng Huang and Zhou Zhang), Communications in Contemporary Mathematics, Vol. 22, No. 07 (2020); arXiv link.

[9] Modified mean curvature flow of entire locally Lipschitz radial graphs in hyperbolic space (joint with Patrick Allmann and Jingyong Zhu), Mathematische Nachrichten, Volume 293, Issue 5, May 2020, 861-878; arXiv link.

[8] Blow-up of the mean curvature at the first singular time of the mean curvature flow (joint with Natasa Sesum), Calculus of Variations and PDEs, June 2016, 55-65; arXiv link.

[7] Stability of the surface area preserving mean curvature flow in Euclidean space (joint with Zheng Huang), Journal of Geometry, December 2015, Volume 106, Issue 3, 483-501; arXiv link.

[6] Uniformity of harmonic map heat flow at infinite time, Analysis & PDEVol. 6 (2013), No. 8, 1899–1921; arXiv link.

[5] Estimates for the energy density of critical points of a class of conformally invariant variational problems (joint with Tobias Lamm), Advances in Calculus of Variations, Volume 6, Issue 4 (Jul 2012), 391--413; arXiv link.

[4] Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space (joint with Ling Xiao), Communications in Analysis and Geometry, Volume 20, Number 5, 1061-1096, 2012; arXiv link.

[3] Closed geodesics in Alexandrov spaces of curvature bounded from above, Journal of Geometric Analysis, Volume 21, Issue 2 (2011), 429-454; arXiv link.

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


Other Publications

[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.