Integrable PDEs and Deligne-Hitchin moduli spaces
Solutions to integrable PDEs on Riemann surfaces like the sinh-Gordon equation or Hitchin's self-duality equations [5] can be described in terms of families of flat connections.
On the one hand, this easily shows that locally, e.g. on the disc, there are (too) many solutions. On the other hand, one can deduce many interesting and deep results for solutions over compact surfaces:
・ for surfaces of genus 0 all solutions can be expressed by holomorphic functions (1980-1990, e.g. Uhlenbeck [11]);
・ for tori, i.e. oriented surfaces of genus 1, one obtains all solutions by means of their algebraic-geometric spectral data (1990-early 2000s, e.g. Hitchin [6]), or again directly by holomorphic functions.
In this course I will explain how to describe solutions of integrable PDEs on compact surfaces of higher genus by means of complex curves into Deligne-Hitchin moduli spaces [2]. In doing so, we will first construct the Deligne-Hitchin moduli space [9,10], and then identify it with the twistor space [7] of solutions of Hitchin's self-duality equations [9]. As a byproduct, we will naturally derive the non-abelian Hodge correspondence [5], see also [4]. After that, we will discuss other equations and their solutions like minimal surfaces in spheres or constant mean curvature surfaces [3]. If time permits, we will discuss certain additional structures on the Deligne-Hitchin moduli space related to hyper-Kähler geometry and to the harmonic map energy [1].
On the one hand, this easily shows that locally, e.g. on the disc, there are (too) many solutions. On the other hand, one can deduce many interesting and deep results for solutions over compact surfaces:
・ for surfaces of genus 0 all solutions can be expressed by holomorphic functions (1980-1990, e.g. Uhlenbeck [11]);
・ for tori, i.e. oriented surfaces of genus 1, one obtains all solutions by means of their algebraic-geometric spectral data (1990-early 2000s, e.g. Hitchin [6]), or again directly by holomorphic functions.
In this course I will explain how to describe solutions of integrable PDEs on compact surfaces of higher genus by means of complex curves into Deligne-Hitchin moduli spaces [2]. In doing so, we will first construct the Deligne-Hitchin moduli space [9,10], and then identify it with the twistor space [7] of solutions of Hitchin's self-duality equations [9]. As a byproduct, we will naturally derive the non-abelian Hodge correspondence [5], see also [4]. After that, we will discuss other equations and their solutions like minimal surfaces in spheres or constant mean curvature surfaces [3]. If time permits, we will discuss certain additional structures on the Deligne-Hitchin moduli space related to hyper-Kähler geometry and to the harmonic map energy [1].
Lecturer
Date
18th September ~ 12th December, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday | 12:30 - 14:15 | A3-4-301 | ZOOM 01 | 928 682 9093 | BIMSA |
| Tuesday | 09:50 - 11:25 | A3-4-301 | ZOOM 01 | 928 682 9093 | BIMSA |
Prerequisite
It is necessary to have some basic knowledge about differential geometry and Riemann surfaces.
Syllabus
・ flat connections and moduli spaces thereof
・ integrable PDEs on Riemann surfaces
・ construction of the Deligne-Hitchin moduli space
・ twistor spaces for hyper-Kähler manifolds
・ loop group factorisations
・ the non-abelian Hodge correspondence
・ harmonic maps via sections of Deligne Hitchin moduli spaces
・ the hyper-holomorphic line bundle
・ integrable PDEs on Riemann surfaces
・ construction of the Deligne-Hitchin moduli space
・ twistor spaces for hyper-Kähler manifolds
・ loop group factorisations
・ the non-abelian Hodge correspondence
・ harmonic maps via sections of Deligne Hitchin moduli spaces
・ the hyper-holomorphic line bundle
Reference
[1] F. Beck, S. Heller, M. Röser, Energy of sections of Deligne-Hitchin moduli spaces, Math. Annalen 380 (2021), no. 3-4, 1169--1214.
[2] I. Biswas, S. Heller, M. Röser, Real sections of the Deligne-Hitchin moduli space, Comm. Math. Phys., Volume 366, Issue 3 (2019), pages 1099--1133.
[3] L. Heller, S. Heller, M. Traizet, Complete families of embedded high genus CMC surfaces in the 3-sphere (with an appendix by Steven Charlton), https://arxiv.org/abs/2108.10214.pdf.
[4] L. Heller, S. Heller, M. Traizet, Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere, https://arxiv.org/abs/2205.12106.pdf.
[5] N. Hitchin, The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59-126.
[6] N. Hitchin, Harmonic maps from a 2-torus to the 3-sphere. J. Differential Geom. 31 (1990), no. 3, 627--710.
[7] N. Hitchin, A. Karlhede, U. Lindström, and M. Rocek, Hyperkähler Metrics and Supersymmetry, Commun. Math. Phys. 108 (1987), 535--589.
[8] L. Heller, S. Heller, M. Traizet, Complete families of embedded high genus CMC surfaces in the 3-sphere, Preprint: arXiv:2108.10214
[9] C. Simpson, The Hodge filtration on nonabelian cohomology. Algebraic geometry--Santa Cruz 1995, 217--281, Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI, 1997
[10] C. Simpson, The twistor geometry of parabolic structures in rank two, https://arxiv.org/pdf/2110.12300.pdf
[11] K. Uhlenbeck, Harmonic maps into Lie groups (classical solutions of the chiral model), J. Diff. Geom. 19 (1984) 431--452.
[2] I. Biswas, S. Heller, M. Röser, Real sections of the Deligne-Hitchin moduli space, Comm. Math. Phys., Volume 366, Issue 3 (2019), pages 1099--1133.
[3] L. Heller, S. Heller, M. Traizet, Complete families of embedded high genus CMC surfaces in the 3-sphere (with an appendix by Steven Charlton), https://arxiv.org/abs/2108.10214.pdf.
[4] L. Heller, S. Heller, M. Traizet, Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere, https://arxiv.org/abs/2205.12106.pdf.
[5] N. Hitchin, The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59-126.
[6] N. Hitchin, Harmonic maps from a 2-torus to the 3-sphere. J. Differential Geom. 31 (1990), no. 3, 627--710.
[7] N. Hitchin, A. Karlhede, U. Lindström, and M. Rocek, Hyperkähler Metrics and Supersymmetry, Commun. Math. Phys. 108 (1987), 535--589.
[8] L. Heller, S. Heller, M. Traizet, Complete families of embedded high genus CMC surfaces in the 3-sphere, Preprint: arXiv:2108.10214
[9] C. Simpson, The Hodge filtration on nonabelian cohomology. Algebraic geometry--Santa Cruz 1995, 217--281, Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI, 1997
[10] C. Simpson, The twistor geometry of parabolic structures in rank two, https://arxiv.org/pdf/2110.12300.pdf
[11] K. Uhlenbeck, Harmonic maps into Lie groups (classical solutions of the chiral model), J. Diff. Geom. 19 (1984) 431--452.
Video Public
Yes
Notes Public
Yes
Lecturer Intro
PhD in 2008, Humboldt Universität Berlin, Germany. Habilitation in 2014, Universität Tübingen, Germany. Professor at Beijing Institute of Mathematical Sciences and Applications since 2022. Research interests: minimal surfaces, harmonic maps, Riemann surfaces, Higgs bundles, moduli spaces, visualisation and experimental mathematics.