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.
Published Journal Articles
[16] Uniqueness of conformal-harmonic maps on locally conformally flat 4-manifolds (joint with Jingyong Zhu), Calculus of Variations and PDEs, Volume 64, article number 57, (2025), arXiv link.
[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 & PDE, Vol. 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.