Research Interests:

My research interest lies at the intersection of differential geometry and partial differential equations, as well as their physical applications, for example, in general relativity and phase transition theory. More precisely, my research focuses on the study of singularities arising in minimal surfaces and mean curvature flow, especially in geometric free boundary problems. Much of my work combines techniques from geometry, partial differential equations and geometric measure theory to study various natural variational boundary value problems arising in geometry and physics.

Preprints / Work in progress

• Boundary behavior of limit-interfaces for the Allen-Cahn equation on Riemannian manifolds with Neumann boundary condition (joint work with D. Parise and L. Sarnataro) arXiv version

• Strong maximum principle for Lipschitz solutions to minimal surface system (joint work with Y. Wang) - in preparation -

• On the existence of min-max minimal free boundary annulus (joint work with J. H.-T. Leung) - in preparation -

• A mixed boundary value problem for Jang's equation and the existence of free boundary marginally outer trapped surfaces (joint work with X. Chai) - in preparation -

Publications in refereed journals

1. Contracting convex surfaces by mean curvature flow with free boundary on convex barriers (joint work with S. Hirsch)     
   Asian J. Math. 27 (2023), no. 2, 187-220   journal link

2. A maximum principle for free boundary minimal varieties of arbitrary codimension (joint work with X. Zhou)
   Comm. Anal. Geom. 29 (2021), no. 6, 1509-1521          journal link

3. Min-max theory for free boundary minimal hypersurfaces I: regularity theory (joint work with X. Zhou)
   J. Differential Geom. 118 (2021), no.3, 487-553           journal link

4. Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disk (joint work with N. Kapouleas)
   J. Reine Angew. Math. 776 (2021), 201-254                  journal link

5. Min-max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications (joint work with Q. Guang, Z. Wang and X. Zhou)
   Math. Ann. 379 (2021), 1395-1424                                  journal link

6. Curvature estimates for stable free boundary minimal hypersurfaces (joint work with Q. Guang and X. Zhou)
   J. Reine Angew. Math. 759 (2020), 245-264                 journal link

7. Chord shortening flow and a theorem of Lusternik and Schnirelmann
   Pacific Journal of Math. 299 (2019), no. 2, 469-488   journal link

8. A general existence theorem of embedded minimal surfaces with free boundary
   Comm. Pure Appl. Math. 68 (2015), no. 2, 286-331     journal link

9. Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary (joint work with A. Fraser)
   J. Differential Geom. 96 (2014), no. 2, 183-200            journal link

10. A sharp comparison theorem for compact manifolds with mean convex boundary
    J. Geom. Anal. 24 (2014), no. 3, 1490-1496                     journal link

11. On complete stable minimal surfaces in 4-manifolds with positive isotropic curvature
    Proc. Amer. Math. Soc. 140 (2012), 2843-2854             journal link

Research reports and proceedings

• Free boundary minimal surfaces in the unit ball: recent advances and open questions
  Proceedings of the International Consortium of Chinese Mathematicians, 2017 (First Annual Meeting), p.401-436, International Press of Boston, Inc. (2020) 654pp. journal link                          arXiv version

• On minimal surfaces with free boundary (joint work with X. Zhou)        pdf

Major Research Grants:

• RGC General Research Fund (2024/25)
  Project: Boundary effects of Ricci curvature and convergence theory of Riemannian manifolds - HK$ 910,742

• RGC General Research Fund (2023/24)
  Project: Geometric flow methods for free boundary minimal surfaces and harmornic maps - HK$ 877,079

• RGC General Research Fund (2022/23)
  Project: Geometric problems in mathematical general relativity - HK$ 1,092,000 

• RGC General Research Fund (2021/22)
  Project: Singularity analysis in minimal surface theory - HK$ 486,015 

• NSFC Excellent Young Scientist Fund (Hong Kong and Macao) (2020)
  Project: Geometric Analysis - RMB$ 1,200,000 

• RGC General Research Fund (2020/21)
  Project: On higher-order isoperimetric inequality - HK$ 599,861 

• RGC General Research Fund (2019/20)
  Project: Geometric variational problems and nonlinear partial differential equations - HK$ 332,261

• RGC General Research Fund (2016/17)
  Project: Global questions in differential geometry on holographic principles - HK$ 488,501

• RGC Early Career Scheme (2015/16)
  Project: Min-max constructions in geometry and applications - HK$ 751,690 

Selected Talk Recordings: 

• Minimal surfaces in S^3  Youtube video
  Minimal surfaces in B^3  Youtube video
  An Invitation to Geometric Analysis (0th Geometric Analysis Festival)

• Mean curvature flow with free boundary   Youtube video
  Asia-Pacific Analysis and PDE Seminar (22 June 2020)