Research

We study the moduli space of all maximal surfaces in pseudo-hyperbolic space H^{2,2} with asymptotic boundary in the Einstein universe Ein^{1,2} that is a polygon of size n+4, up the action of SO_0(2,3). Maximal surfaces generally have an associated holomorphic quartic differential, and these polygonal surfaces are characterized by being conformally planar and having polynomial quartic differential. Importantly, fixing the polygons to have n+4 vertices, the resulting moduli space M_{n+4}(H^{2,2}) is disconnected when n is at least two.  We introduce a turning invariant that labels the connected components. In fact, this turning invariant is defined using the connection between Ein^{1,2} and the sphere S^{2}, and so leads to a connection with spherical polygons. We parametrize the moduli space M_{n+4}(H^{2,2}) in terms of holomorphic data related to the complexified second fundamental form of the surfaces, namely a pair of polynomials. We then relate these polynomials to the turning invariant. As a consequence, we obtain a geometric characterization of the Hitchin component in the moduli space, which was first studied by Tamburelli-Wolf.

We continue the quest for explicit geometric structures interpretations of Hitchin components, extending the 'geometrization' of surface group representations exhibited by Teichmüller space. In particular, we consider the line-hyperplane structures associated to PSL(n, R)-Hitchin representations by work of Guichard-Wienhard. We lift the manifold M carrying these line-hyperplane structures to a four-fold cover, which is a fiber bundle over the original surface S, and study the lifted fiber using techniques from high-dimensional differential topology. More specifically, using Wall's classification of highly connected odd-dimensional manifolds via surgery theory, we determine the almost diffeomorphism type of the fiber of the cover.

Let S be a closed surface of genus at least two. We construct geometric structures modeled on the two G_2'-partial flag on closed 5-manifolds that fiber over S using Higgs bundles. In particular, we consider two families of cyclic Higgs bundles that we call 𝛼-bundles and β-bundles. Here, 𝛼, β are the long and short roots, respectively, in the G2-root system. From Higgs bundles in the first family that are Hodge bundles, we build geometric structures in the flag manifold of the long root, and from Higgs bundles in the second family, we build geometric structures in the flag manifold of the short root. This construction provides two large families of such (G,X)-manifolds, each of which is stratified. We note that the geometric structures of type 𝛼 are rather 'exotic', hard to build, and seem to be not very well-studied. For those of type β, we obtain a corollary of interest. Namely, we give an explicit description of the geometric structures defined by Guichard-Wienhard in the Einstein universe Ein^{2,3}, the flag manifold of the short root, for G_2'-Hitchin representations. A priori, these geometric structures were defined with completely different techniques, namely as compact quotients of domains of discontinuity. Our construction also gives a unified perspective on past analytic approaches to build geometric structures by many authors, which we describe in detail in an appendix. 

We study geometric structures in the Einstein Universe Ein^{p-1,p} for representations of closed surface groups in G = SO(p,p+1) for a large class of Anosov representations that includes Hitchin representations . These representations admit an open and cocompact domain of discontinuity in Ein^{p-1,p}, whose quotient M is known to fiber over the original surface S as an F-fiber bundle. We explicitly determine this fiber F, and show that in many cases, the global fiber bundle F --> M --> S is actually trivial. The main idea is to describe the fiber F as a base of pencil, using Busemann functions and the geometry of the G-symmetric space, and then use Higgs bundles to assemble a global "moving base of pencil" fiber bundle, modeled pointwise on this description, that is diffeomorphic to the quotient M.  

We study the notion of transversality in flag manifolds, inpired by the relation with Anosov representations of hyperbolic groups. Recall a subset of a subset of a (self-opposite) flag manifold is called transverse when every pair of distinct points is in "general position".  For some partial flag manifolds of semisimple real Lie groups, including many full flag manifolds, transverse circles are known to be locally maximally transverse. We complete the classification of all partial flag manifolds of split real Lie groups with this property. 

On the other hand, using spinors, we find the first higher-dimensional examples of transverse spheres in full flag manifolds of split real groups, including many of classical types An, Bn, Dn, depending on parity of n modulo 8, and of exceptional types G2, F4, E6, E8. The new transverse spheres obtain arbitrarily large dimensions. Many of these transverse spheres are verified to be homotopically nontrivial and hence maximally transverse with tools from topological K-theory, namely the remarkable work of Atiyah-Bott-Shapiro that relates stable properties of vector bundles on spheres to representation theory of Clifford algebras (and hence spinors). The aforementioned classification follows from constructions of transverse m-spheres, m at least two, that complement the previously known restrictions as well as a new restriction in the type E7 case. Additionally, when G is split of type G2, B3, or D4, the full flag manifold admits a fibration by maximally transverse 3-spheres.

We give an explicit geometric structures interpretation of the G_2'-Hitchin component. This adds a new concrete case to the short list of Lie groups for which such a result was previosly known, namely PSL(2,R), SL(3,R), SO_0(2,3) = PSp(4,R), PSL(4,R). The philosophy of the construction is similar to in previous work by Baraglia and Collier-Tholozan-Toulisse, namely to use associated equivariant harmonic maps. In this case, we use the equivariant alternating almost-complex curves in S^{2,4} introduced by Baraglia, and later proved to be canonically attached to Hitchin representations by work of Labourie and Collier-Toulisse. Using the geometry of the Frenet frame of such curves, we convert such curves into non-compact 5-manifolds that fiber over the original surface S and carry a geometric structure in the Einstein universe Ein^{2,3}. The fiber F of this 5-manifold is S^1 x S^ 1 x R. We then build an appropriate moduli space of 'decorated' geometric strutures, and prove this moduli space is canonically homeomorphic to the G_2'-Hitchin component. It is noteworthy that the construction does not use domains of discontinuity, but intead differential geometry. We study this construction concretely in Fuchsian-Hitchin case in detail. Some strange and pathological features arise: the developing map is non-proper, points in the image have 1,2, or 3 pre-images [each case does arise], and dev is highly non-injective: the developing map is never injective on any open neighborhood of a single fiber. As a consequence, these structures are not related to those defined by Guichard-Wienhard.

We study almost-complex curves in the pseudosphere S^{2,4}, the unit sphere of pseudo-Euclidean space R^{3,4}. The space R^{3,4} is quite special: it is one of few pseudo-Euclidean spaces to admit a cross-product as in familiar Euclidean space E^3. The cross-product on R^{3,4} is invariant by the split real Lie group G_2' (in fact, G_2' can be characterized as its stabilizer) and this cross-product leads to the canonical (non-integrable) almost-complex structure on S^{2,4}, allowing us to study almost-complex curves from a Riemann surface. One can imagine S^{2,4} as the non-compact analogue of S^6 with its canonical almost complex structure coming from Euclidean 7-space, which is invariant by the compact real form of G_2. 

The almost-complex curves of interest arise from the g_2 affine Toda equations, or equivialently the Hitchin section of Higgs bundles over the complex plane, associated to a holomorphic sextic differential q_6. We study the case that q_6 is a polynomial. The resulting almost-complex curves have interesting geometric features: they are strictly negatively curved, asymptotically flat, and have a polygonal asymptotic boundary in the Einstein universe Ein^{2,3}, the frontier of the projective pseudosphere in RP^6. We show these boundary polygons have a G_2'-geometric feature of interest called the annihilator property, which can be characterized as the edges of the polygon being integral curves of a maximally non-integrable 2-plane distrubtion D on Ein^{2,3}, also called a [2,3,5]-distribution, first described by Cartan.