Schedule for: 18w5198 - Higgs Bundles and Harmonic Maps of Riemann Surfaces

After the work of Hitchin, Simpson and Biquard-Boalch, we know the correspondence between irreducible wild harmonic bundles and stable good filtered Higgs bundles of degree 0. It particularly provides us with a powerful method to study twisted harmonic maps on punctured Riemann surfaces. In this talk, we shall explain how we could apply this theory to prove the existence and the uniqueness of solutions for Toda type equations on punctured compact Riemann surfaces. This is one of the two methods developed in a joint work with Qiongling Li.

Moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. For representations into complex semisimple Lie groups, the components of these spaces are labeled by obvious topological invariants. This is no longer true if one restricts to real forms of the complex groups. Factors other than the obvious invariants lead to the existence of extra `exotic' components which can have special significance. All formerly known instances of such exotic components were attributable to one of two distinct mechanisms and collectively comprise so-called ``higher Teichmuller'' components. Recent Higgs bundle results for the groups SO(p,q) shed new light on these phenomena and reveal new examples outside the scope of the two known mechanisms. This talk will describe the new SO(p,q) results.

Parreau compactified the Hitchin component of a closed surface of negative Euler characteristic in such a way that a boundary point corresponds to the projectivized length spectrum of an action on a Euclidean building. In this talk, we use the positivity properties of Hitchin representations introduced by Fock and Goncharov to explicitly describe the geometry of a preferred collection of apartments in the limiting building.

The moduli space of Higgs bundles on a Riemann surface is a (usually singular) hyperk\"ahler manifold. Its Twistor space is called the Deligne-Hitchin moduli space, which is a complex space that fibres over the complex projective line. Solutions to Hitchin's self-duality equations give rise to holomorphic sections of this fibration satisfying a certain reality condition. A natural question, asked by Simpson, is whether any such "real" section comes from a solution to the self-duality equations. There are other natural reality conditions one can impose, leading to different types of real holomorphic sections of the Deligne-Hitchin Twistor space. In this talk we will describe their relation to various kinds harmonic maps from the Riemann surface into homogeneous spaces and, time permitting, give an answer to Simpson's question. This is joint work with I.Biswas and S.Heller.

A general problem in Teichmuller theory consists in studying canonical maps between hyperbolic surfaces. When the surfaces are closed, minimal Lagrangian maps appear naturally as minimisers of the holomorphic one-energy and their properties are well-understood thanks to the work of Labourie and Schoen. In this talk, I will use the correspondence between minimal Lagrangian maps and maximal surfaces in anti- de Sitter space to construct minimal Lagrangian maps between surfaces with geodesic boundary and cusps.

Many problems in classical geometry are related to harmonic maps via some kind of Gauss map (CMC surfaces, affine spheres, Willmore surfaces, projectively minimal surfaces...). In many cases, this relates an intractable parabolic geometry (conformal, projective...) with the more familiar (pseudo-)Riemannian geometry of a symmetric space for which integrable systems methods are available. In this talk, I will describe a unified approach to some of these problems via the very classical notion of a line congruence in projective geometry and its Laplace invariant.

I will report on recent joint work with Rafe Mazzeo, Jan Swoboda and Frederik Witt on the asymptotic geometry of the Hitchin metric. This is the natural metric on the moduli space of Higgs bundles. We describe the difference to a more elementary semiflat metric, thus confirming part of a more general proposal of Gaiotto, Moore and Neitzke.

Minkowski space of dimension 2+1 is the Lorentzian analogue of Euclidean 3-space. It is well-known that the Gauss map of a Riemannian surface of constant mean curvature (CMC) in Minkowski space is harmonic, while the Gauss map of a surface of constant Gaussian curvature (CGC) is minimal Lagrangian. In this talk I will present a classification result for properly embedded CMC and CGC surfaces in Minkowski space, and show how harmonic maps from the complex plane to the hyperbolic plane play an essential role in the proof. This is joint work with Francesco Bonsante and Peter Smillie.

A variation of twistor structures in the sense of Simpson, Sabbah and Mochizuki is a 1-parameter family of flat connections on a complex vector bundle with (to be chosen) additional data and constraints. Some version on rank 2 bundles turns up in the DPW method for constructing CMC surfaces. Another version of arbitrary rank is equivalent to Simpson's harmonic bundles, which are a generalization and weakening of variation of Hodge structures. A generalization of variation of Hodge structures which is a not a weakening, can be encoded as an integrable variation of twistor structures. Closely related versions of this are tt^* geometry (Cecotti-Vafa), TERP structures (Hertling) and noncommutative Hodge structures (Katzarkov-Kontsevich-Pantev). In the talk, I will discuss these structures in general, and in distinguished cases which arise in the theory of isolated hypersurface singularities. A good way to control them is given by the theory of meromorphic connections with irregular poles and their Stokes structures. I will sketch some results and some open questions.

The Teichmüller space T of a closed surface S of genus greater than or equal to two is the space of marked hyperbolic structures on S. In the 1980s M.Wolf and N. Hitchin independently showed that T can be parametrized by holomorphic quadratic differentials, with respect to a choice of complex structure, on S. I shall describe how this parametrization can be extended to a correspondence between meromorphic quadratic differentials with higher order poles, and the Teichmüller space of crowned hyperbolic surfaces. The proof involves showing the existence of a harmonic map from a punctured Riemann surface to a crowned hyperbolic surface, with prescribed principal parts of its Hopf differential, which determine the asymptotic behaviour of the map near the punctures. I shall also describe how such a map arises as a limit of harmonic maps as the domain Riemann surface degenerates along a Teichmüller ray.

I will talk about a work in progress in collaboration with Christian El Emam, whose final motivation is to study the geometry of surface group actions on SL(2, C) regarded as the homogeneous space SO(4,C)/SO(3, C). I will mainly focussed on some analytical issues related to equivariant immersions. We will introduce the notion of first and second fundamental forms for an immersion and prove that they are solutions of a complex version of the standard Gauss Codazzi equation in the hyperbolic setting. We will discuss how far this theory can be regarded as a complex version of the Anti de Sitter geometry . In particular we will introduce a notion of left/right Gauss maps for an immersion that extends the corresponding notions in the Anti de Sitter setting. In the final part of the talk we will introduce the notion of minimal immersion in this context and will try to give a general description of the embedding data of minimal surfaces in terms of holomorphic objects.

Moduli spaces of stable vector bundles carry a natural Kähler structure, described originally in the Riemann surface case by Narasimhan and in the pioneering work of Atiyah-Bott. Such a Kähler structure is in many ways analogous to the Weil-Petersson metric on moduli spaces of Riemann surfaces, for which a deep relationship with the Liouville functional in Conformal Field Theory was established by Takhtajan and Zograf. In this talk I will describe work in progress on how the ideas of Takhtajan-Zograf can be adapted to vector bundles in three different settings: moduli of stable parabolic bundles in genus 0 and 1, moduli of semistable bundles in genus 1, and Jacobians. In all cases the main tool is an adaptation of the WZNW action of Conformal Field Theory---defined by twisting the so-called chiral models with a topological term---to a functional on singular hermitian metrics on a suitable holomorphic gauge. I will also describe briefly how the previous results can be generalized to moduli spaces of parabolic Higgs bundles.

Solutions of Hitchin's self-duality equations correspond to special real sections in the Deligne-Hitchin moduli space -- twistor lines. A question posed by Simpson in 1995 asks whether all real sections give rise to global solutions of the self-duality equations. An armative answer would allow for complex analytic procedure to obtain solutions of the self- duality equations. The purpose of my talks is to explain the construction of counter examples given by certain (branched) Willmore surfaces in 3-space (with monodromy) via the generalized Whitham flow. Though these higher solutions do not give rise to global solutions of the self- duality equations on the whole Riemann surface M, they are solutions on an open dense subset of it. This suggest a deeper connection between Willmore surfaces, i.e., a rank 4 harmonic map theory, with the rank 2 self-duality theory.

Both the Higgs bundle moduli space and the moduli space of flat connections have a natural stratification induced by a C* action. In both of these stratifications, each strata is a holomorphic fibration over a connected component of complex variations of Hodge structure. While the nonabelian Hodge correspondence provides a homeomorphism between Higgs bundles and flat connections, this homeomorphism does not preserve the respective strata. The closed strata on the Higgs bundle side is the image of the Hitchin section (the Hitchin component) and the closed strata in the space of flat connections is the space of opers. In this talk we will show how many of the relationships between opers and the Hitchin component extend to general strata. This is based on joint work with Richard Wentworth.

In their study of certain two-dimensional physical theories, Cecotti and Vafa discovered the tt*-equations. These are equations in terms of bundle data over the moduli spaces of these theories and their solutions are referred to as tt*-geometry. In this talk, based on joint work with Murad Alim and Laura Fredrickson, we study a particular class of tt*-geometry and match the tt*-equations with Hitchin’s equations. At the boundary of the corresponding moduli spaces of theories, parabolic structures naturally appear and we determine them explicitly in a wide range of examples. Finally, we comment on an oper limit of Hitchin’s equations in the parabolic framework.

The moduli space of SL(n,C)-Higgs bundles over a Riemann surface admits a Hitchin fibration over the space of holomorphic differentials on the Riemann surface. In this talk, we focus on Hitchin fibers at (q2,0,...,0), that is, the fibers containing Fuchsian locus. We show a comparison theorem on the length spectrum between surface group representations in such fibers with Fuchsian ones, as a generalization of the SL(2,C) case shown by Deroin and Tholozan.

The pseudo-hyperbolic space $\mathbb{H}^{p,q}$ is a pseudo-Riemannian analogue of the classical hyperbolic space. In this talk, I will explain how equivariant maximal surfaces in $\mathbb{H}^{2,n}$ are related to harmonic maps into some non-compact symmetric space (so in particular to Higgs bundles). This link provides a powerful tool to study maximal representations in rank 2 Hermitian Lie groups. This is a joint work with Brian Collier and Nicolas Tholozan.

A triangulation of the sphere is combinatorially convex if each vertex is shared by no more than six triangles. In joint work with Philip Engel, we show that counted appropriately, the number of triangulations of the sphere with $2n$ triangles is the $n$th Fourier coefficient of a certain multiple of the Eisenstein series $E_{10}$. Our method is based on Thurston's description of triangulations as lattice points in a stratum of sextic differentials. It generalizes in a straightforward way to show that the number of convex tilings of a sphere by squares or by hexagons also form the coefficients of a modular form. As a consequence, we reproduce formulas for Masur-Veech volumes of certain strata of cubic, quartic, and sextic differentials. Time permitting, I will describe an approach to counting problems in strata of differentials of all orders.

We will recall the general loop group technique for primitive harmonic maps from Riemann surfaces to k-symmetric spaces and will give more concrete applications to such maps associated with the Kac-Moody algebra $A_2^2$.