Research

Research Interests

      Higher Teichmueller Theory, Higgs Bundles, and Harmonic Maps

Papers

1. Complete solutions of Toda equations and cyclic Higgs bundles over non-compact surfaces, with Takuro Mochizuki, accepted in International Mathematics Research Notices,  arXiv:2010.05401.

2. Higgs bundles in the Hitchin section over non-compact hyperbolic surfaces, with Takuro Mochizuki, accepted in Proceedings of the London Mathematical Society, arXiv:2307.03365

3. Projective structures with (Quasi-)Hitchin holonomy, with Daniele Alessandrini, Colin Davalo, accepted in Journal of the London Mathematical Society,  arXiv:2110.15407.

4. Bounded differentials on the unit disk and the associated geometry, with Song Dai, Transactions of the American Mathematical Society, Volume 377, Number 8, August 2024, Pages 5445–5481 .  

5. Harmonic metrics of generically regular semisimple Higgs bundles on non-compact Riemann surfaces, with Takuro Mochizuki, Tunisian Journal of Mathematics, Vol. 5 (2023), No. 4, 663–711.

6. Domination results in n-Fuchsian fibers in the moduli space of Higgs bundles, with Song Dai,  Proceedings of the London Mathematical Society,Volume124, Issue 4, April 2022, Pages 427-477, arXiv:2005.13960.

7.  Nilpotent Higgs bundles and the Hodge metric on the Calabi-Yau moduli, International Mathematics Research Notices, Volume 2022, Issue 9, May 2022, Pages 6705–6741,  arXiv:2005.13939.

8. An introduction to Higgs bundles via harmonic maps, SIGMA 15 (2019), 035, 30 pages. 

9. Harmonic maps for Hitchin representations, Geometric and Functional Analysis, (2019) 29 (2), 539-560.

10. On cyclic Higgs bundles, with Song Dai, Mathematische Annalen volume 376, pp. 1225–1260(2020), arXiv:1710.10725.

11. On the uniqueness of vortex equations and its geometric applications, The Journal of Geometric Analysis (2019) 29:105–120.

12. Minimal surfaces for Hitchin representations, with Song Dai,  Journal of Differential Geometry, Vol. 112, No. 1 (2019), pp. 47-77, arXiv:1605.09596.

13. AdS 3-manifolds and Higgs bundles, with Daniele Alessandrini, Proc. Amer. Math. Soc. 146 (2018), pp. 845-860, arXiv:1510.07745.

14. Asymptotics of Higgs bundles in Hitchin Components, with Brian Collier, Advances in Mathematics, Vol 307 (2017), pp. 488–558, arXiv:1405.116.

15. Teichmueller space is totally geodesic in Goldman space, Asian Journal of Mathematics, Vol 20 (2016) No. 1, pp. 21-46.

Preprints

16. Harmonic metrics of generically regular nilpotent Higgs bundles over non-compact surfaces, with Song Dai, arXiv:2412.14429

17. Every closed surface of genus at least 18 is Loewner, with Weixu Su, arXiv:2401.00720

18. Isolated singularities of Toda equations and cyclic Higgs bundles, with Takuro Mochizuki, arXiv:2010.06129.

Notes

• Lecture notes on metric geometry of Euclidean buildings. These are the note from an introductory talk at a GEAR log cabin Workshop on Compactifications of moduli spaces of representations.  (full notes from the workshop)

• slides for the talk on "Cyclic Higgs bundles and minimal surfaces", Conference of Complex Analysis in China, Qufu, 2017

• slides for the talk on "Asymptotics of certain families of Higgs bundles", AMS-EMS-SPM International Meeting in Porto, 2015

• slides for the talk on "Teichmueller space is totally geodesic in Goldman space", Joint Mathematics Meetings in San Diego, 2013.

• Lectures notes on Convex RP^2 structures. These are the note from a talk I gave at a GEAR log cabin Workshop on Higher Teichmueller-Thurston Theory.

Conference (co-organized)

• 2020 Frontier of Analysis and Geometry of Teichmüller spaces, TSIMF, Sanya, Jan 06-10, 2020. 

• Workshop on Teichmüller theory and Related topics, Tsinghua University, Beijing, Aug 24-26, 2018.  

• Workshop on Surface Group Representations, Mar 17-20, 2016.

• Workshop on Higgs bundles and Harmonic Maps, Jan 3-10, 2015.