My research focuses on fully nonlinear equations arising from geometric problems and related geometric flow problems. I study existence, uniqueness, regularity, and qualitative properties of solutions. I am especially interested in the interaction between these two fields, that is, using fully nonlinear equations to study curvature flows or using flow methods to solve classical problems in PDE.

Publications

19.Wang, Zhizhang; Xiao, Ling Entire spacelike hypersurfaces with constant $\sigma_{n-1}$ curvature in Minkowski space.

     To appear on  J. Geom. Anal.

18. Wang, Zhizhang; Xiao, Ling Entire convex curvature flow in Minkowski space.

      Calc. Var. Partial Differential Equations 62 (2023), no. 9, Paper No. 252, 27 pp. 

17. Ren, Changyu; Wang, Zhizhang; Xiao, Ling The prescribed curvature problem for entire hypersurfaces in Minkowski space. 

       Anal. PDE 17 (2024), no. 1, 1-40. 

16. Xiao, Ling  Asymptotic convergence for modified scalar curvature flow. 

      Comm. Anal. Geom. 31 (2023), no. 1, 69–96. 

15. Wang, Zhizhang; Xiao, Ling Entire self-expanders for power of $\sigma_k$ curvature flow in Minkowski space.

      J. Funct. Anal. 284 (2023), no.8, Paper No. 109866, 27 pp.

14. Wang, Zhizhang; Xiao, Ling Entire spacelike hypersurfaces with constant $\sigma_k$ curvature in Minkowski space.

      Math. Ann. 382 (2022), no.3-4, 1279–1322.

13. Xiao, Ling Neumann boundary value problem for general curvature flow with forcing term.

      Geom. Dedicata 213 (2021), 345–358.

12. Jiang, Xumin; Xiao, Ling Free boundary regularity on the focusing problem for the  Qk curvature flow with flat sides I.

      J. Funct. Anal. 280 (2021), no.2, Paper No. 108792, 37 pp.

11. Spruck, Joel; Xiao, Ling Complete translating solitons to the mean curvature flow in $R^3$ with nonnegative mean curvature. 

         Amer. J. Math. 142 (2020), no.3, 993–1015.

10. McGonagle, Matthew; Xiao, Ling Minimal graphs and graphical mean curvature flow in $M^n \times R$. 

      J. Geom. Anal. 30 (2020), no.2, 2189–2224.

9. Jiang, Xumin; Xiao, Ling Optimal regularity of constant curvature graphs in hyperbolic space. 

    Calc. Var. Partial Differential Equations 58 (2019), no.4, Paper No. 133, 22 pp.

8. Spruck, Joel; Xiao, Ling A note on star-shaped compact hypersurfaces with prescribed scalar curvature in space forms. 

    Rev. Mat. Iberoam. 33 (2017), no.2, 547–554.

7. Spruck, Joel; Xiao, Ling Entire downward translating solitons to the mean curvature flow in Minkowski space. 

    Proc. Amer. Math. Soc.144 (2016), no.8, 3517–3526.

6. Chang, Jui-En; Xiao, Ling The Weyl problem with nonnegative Gauss curvature in hyperbolic space. 

      Canad. J. Math. 67 (2015), no.1, 107–131.

5. Guan, Bo; Spruck, Joel; Xiao, Ling Interior curvature estimates and the asymptotic plateau problem in hyperbolic space. 

      J. Differential Geom. 96 (2014), no.2, 201–222.

4. Xiao, Ling Curvature flow of complete convex hypersurfaces in hyperbolic space. 

    J. Geom. Anal. 23 (2013), no.4, 1641–1673.

3. Xiao, Ling  Curvature flow of complete hypersurfaces in hyperbolic space.  

    Geom. Dedicata 164 (2013), 357–383. 

2. Lin, Longzhi; Xiao, Ling Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space. 

    Comm. Anal. Geom. 20 (2012), no.  5, 1061–1096.

1. Spruck, Joel; Xiao, Ling  Convex spacelike hypersurfaces of constant curvature in de Sitter space. 

    Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no. 6, 2225–2242. 

Preprints

20. Wang, Zhizhang; Xiao, Ling Entire spacelike constant $\sigma_k$ curvature hypersurfaces with prescribed boundary data at infinity.

21. Wang, Zhizhang; Xiao, Ling Entire $\sigma_k$ curvature flow in Minkowski space. 

22. Xiao, Ling Generalized Minkowski inequality via degenerate Hessian equations on exterior domains.

23. Xiao, Ling On the exterior Dirichlet problem for Hessian equations in Hyperbolic space. Available upon request.

24. Wang, Zhizhang; Xiao, Ling Non-convexity of level sets for k-Hessian equations in convex ring.