Research

My research lies in the field of Global Surface Geometry. I am investigating constrained Willmore tori, i.e., critical points of the Willmore energy with prescribed conformal class, and CMC surfaces (of higher genus) using the Integrable Systems approach, where moduli spaces of flat connections naturally appear.
I am particularly interested in the interplay between Algebraic Geometric data of the surfaces coming from Integrable Systems (e.g., Higgs bundles, spectral curves) and their analytic properties (e.g., non degeneracy and stability) studied in Geometric Analysis.

Experimental minimizers of Willmore energy with prescribed conformal class. Images by Nick Schmitt .

Publications

Preprints

1. Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere

   with Lynn Heller and Martin Traizet
   Preprint: arxiv: 2205.12106

2. Fuchsian DPW potentials for Lawson surfaces

   with Lynn Heller
   Preprint: arxiv: 2202.05184

3. On the existence of holomorphic curves in compact quotients of SL(2,C)

   with Indranil Biswas, Sorin Dumitrescu and Lynn Heller
   Preprint: arxiv: 2112.03131

4. Complete families of embedded high genus CMC surfaces in the 3-sphere

   with Lynn Heller and Martin Traizet
   Preprint: arxiv: 2108.10214

5. Holomorphic sl(2,C)-systems with Fuchsian monodromy (with an appendix by Takuro Mochizuki)

   with Indranil Biswas, Sorin Dumitrescu and Lynn Heller
   Preprint: arxiv: 2104.04818

6. Exploring the Space of Compact Symmetric CMC Surfaces.

   With Sebastian Heller and Nicholas Schmitt
   Preprint: arXiv:1503.07838
   
   
   

Accepted and published

1. Area estimates for high genus Lawson surfaces via DPW

   with Lynn Heller and Martin Traizet
   accepted for publication in Journal of Differential Geometry
   Preprint: arxiv: 1907.07139
   
   

2. First explicit constrained Willmore minimizers of non-rectangular conformal class.

   Adv. Math., volume 386, paper no. 107804, 2021 With Cheikh Birahim Ndiaye
   Preprint: arXiv:1710.00533
   
   

3. Candidates for non-rectangular constrained Willmore minimizers

   With Cheikh Birahim Ndiaye
   J. Geom. Phys., volume 165, paper no. 104221, 2021
   Preprint: arXiv:1902.09572
   
   

4. Stability properties of 2-lobed Delaunay tori in the 3-sphere

   With Sebastian Heller and Cheikh Birahim Ndiaye
   Differ. Geom. Phys., volume 79, paper no. 101805, 2021
   Preprint: arXiv:1903.11830
   
   

5. Isothermic constrained Willmore tori in the 3-space

   With Sebastian Heller and Cheikh Birahim Ndiaye
   Ann. Glob. Anal. Geom., volume 60, pp 231--251,2021
   Preprint: arXiv:1903.11823
   
   

6. Generalized Whitham flow and its applications

   Minimal surfaces: Integrable Systems and Visualization, Springer Proceedings in Mathematics and Statistics 348, pp 131--146, 2021
   
   

7. Higher solutions of Hitchin's self-duality equations

   With Sebastian Heller
   J. Int. Syst., volume 5, no.1, xyaa006, 2020
   Preprint: arXiv:1801.02402.
   
   

8. Navigating the Space of Symmetric CMC Surfaces

   With Sebastian Heller and Nicholas Schmitt
   J. Differ. Geom., volume 110, no. 3, pp413--455, 2018
   Preprint: arXiv:1501.01929
   
   

9. Towards a constrained Willmore conjecture

   With Franz Pedit
   Willmore energy and Willmore conjecture, pp 119--138, Monogr. Res. Notes Math., CRC Press, Boca Raton, FL, 2018
   Preprint: arXiv:1705.03217



10. Dirac Tori

    Differ. Geom. Appl., vol. 54, Part A, pp. 122-128, 2017
    Preprint: arXiv:1401.7449



11. The spectral curve theory for (k,l)-symmetric CMC surfaces

    With Sebastian Heller and Nicholas Schmitt
    J. Geom. Phys., vol 98, pp 201-213, 2015
    Preprint: arXiv:1503.00969



12. Abelianization of Fuchsian systems on a 4-punctured sphere and applications

    With Sebastian Heller
    J. Symplect. Geom., vol. 14, no. 4, pp. 1059-1088, 2016
    Preprint: arXiv:1404.7707
    
    

13. Constrained Willmore and CMC tori in the 3-sphere

    Differ. Geom. Appl., vol. 40, pp 232-242, 2015.
    Preprint: arXiv:1212.2068



14. Equivariant Constrained Willmore Tori in the 3-Sphere

    Math. Z., vol. 278, no. 3, pp 955-977, 2014
    Preprint: arXiv:1211.4137
    
    

15. Constrained Willmore Tori and Elastic Curves in 2-Dimensional Space Forms.

    Comm. Anal. Geom., vol. 22, no. 2, pp 343-369, 2014
    Preprint: arXiv:1303.1445
    
    

16. Constrained Willmore Hopf tori

    
    Oberwolfach Reports, vol. 10, no. 2., 2013
    
    

17. Equivariant Constrained Willmore Tori in S^3.

    PhD Thesis, Eberhard Karls University Tübingen, 2012.