arXiv subjects

Parker Evans

Publications and source records attributed to Parker Evans.

6 recordsLinked to original sources

arXiv2026

$G_2'$-Slodowy Slices I: Geometric Structures

Let $S$ be a closed surface of genus $g \geq 2$. We construct locally homogeneous geometric structures on closed 5-manifolds fibering over $S$, modeled on the two partial flag manifolds $\mathrm{Ein}^{2,3}$ and $\mathrm{Pho}^\times$ of the split real form $\mathrm{G}_2'$ of the complex exceptional Lie group $\mathrm{G}_2^{\mathbb{C}}$. To this end, we consider two families of representations $π_1S\rightarrow \mathrm{G}_2'$ constructed via the non-abelian Hodge correspondence from cyclic Higgs bundles, one associated with each $\mathrm{G}_2'$-partial flag manifold. Each family includes $\mathrm{G}_2'$-Hitchin representations, but is much more general. From the Higgs bundles of the first family, called $β$-bundles, we construct $(\mathrm{G}_2', \mathrm{Ein}^{2,3})$-geometric structures on $\mathrm{Ein}^{2,1}$-fiber bundles over $S$, and from Hodge bundles in the second family, called $α$-bundles, we construct $(\mathrm{G}_2', \mathrm{Pho}^\times)$-geometric structures on $(\mathbb{RP}^2\times \mathbb{S}^1)$-bundles over $S$. In the case of $\mathrm{G}_2'$-Hitchin Hodge bundles, which belong to both families, we show the image of the developing map of the respective geometric structures is exactly the domain of discontinuity defined by Guichard-Wienhard and Kapovich-Leeb-Porti. Each construction can be interpreted as converting a family of equivariant $J$-holomorphic curves in the pseudosphere $\hat{\mathbb{S}}^{2,4}$ into geometric structures on fiber bundles $M \rightarrow S$. The approach used to build geometric structures, namely \emph{moving bases of pencils}, gives a unified description of prior analytic geometric structures constructions using Higgs bundles and harmonic maps.

math.DG↗

arXiv2026

On Einstein Structures for $\mathrm{SO}_0(p,p+1)$-Surface Group Representations

Let $S$ be a closed surface of genus $g \geq 2$. We study the cocompact domain of discontinuity $Ω_ρ$ in the Einstein universe $\mathrm{Ein}^{p-1,p}$ defined by Guichard-Wienhard and Kapovich-Leeb-Porti for a class of $p$-Anosov representations $ρ:π_1S \rightarrow \mathrm{SO}_0(p,p+1)$ including Hitchin representations, for $p \geq 3$. The quotient $M_ρ = ρ(π_1S)\backslash Ω_ρ$ is abstractly known to be realizable as a fiber bundle over $S$, with unknown fiber of unique homotopy type $F_ρ$. We explicitly exhibit $M_ρ$ as a smooth $\mathfrak{F}_ρ$-fiber bundle over $S$, determining the diffeomorphism type of $M_ρ$ and the unique homotopy type $F_ρ$. Surprisingly, in many situations the fiber bundle $\mathfrak{F}_ρ \rightarrow M_ρ\rightarrow S$ is trivial.

math.GT↗

arXiv2025

Transverse Spheres in Flag Manifolds

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. As a consequence, $\{7\}$-Anosov subgroups of split $E_7$ are virtually free or surface groups. On the other hand, using spinors, we find transverse spheres of arbitrarily large dimension in certain full flag manifolds of Cartan-Killing types $A,B,D$. These transverse spheres are verified to be maximally transverse with tools from topological $K$-theory. The aforementioned classification follows from constructions of transverse $m$-spheres, $m \geq 2$, that complement the previously known restrictions as well as the new $E_7$ restriction. Additionally, when $G$ is split of type $G_2, B_3,$ or $D_4$, the full flag manifold admits a fibration by maximally transverse 3-spheres.

math.GT↗

arXiv2024

Geometric Structures for the $G_2'$-Hitchin Component

We give an explicit geometric structures interpretation of the $G_2'$-Hitchin component $Hit(S, G_2') \subset χ(π_1S,G_2')$ of a closed oriented surface $S$ of genus $g \geq 2$. In particular, we prove $Hit(S, G_2')$ is naturally homeomorphic to a moduli space $\mathscr{M}$ of $(G,X)$-structures for $G = G_2'$ and $X = Ein^{2,3}$ on a fiber bundle $\mathscr{C}$ over $S$ via the descended holonomy map. Explicitly, $\mathscr{C}$ is the direct sum of fiber bundles $\mathscr{C} = UTS \oplus UTS \oplus \underline{\mathbb{R}_+}$ with fiber $\mathscr{C}_p = UT_p S \times UT_p S \times \mathbb{R}_+$, where $UT S$ denotes the unit tangent bundle. The geometric structure associated to a $G_2'$-Hitchin representation $ρ$ is explicitly constructed from the unique associated $ρ$-equivariant alternating almost-complex curve $\hatν: \tilde{S} \rightarrow \hat{\mathbb{S}}^{2,4}$; we critically use recent work of Collier-Toulisse on the moduli space of such curves. Our explicit geometric structures are examined in the $G_2'$-Fuchsian case and shown to be unrelated to the $(G_2', Ein^{2,3})$-structures of Guichard-Wienhard.

math.DG↗

arXiv2024

Polynomial Almost-Complex Curves in $\hat{\mathbb{S}}^{2,4}$

For solutions to the $\mathfrak{g}_2$ affine Toda field equations in $\mathbb{C}$ with respect to \emph{polynomial} holomorphic sextic differential $q$, we study the associated almost-complex curves $ν_q: \mathbb{C} \rightarrow \hat{\mathbb{S}}^{2,4}$. The asymptotic boundary $Δ:= \partial_{\infty}(ν_q)$ of $ν_q$ is found to be a polygon in $\mathsf{Ein}^{2,3}$ with $\mathsf{deg} q + 6$ vertices. The polygon $Δ$ satisfies an \emph{annihilator property}, which is related to a $\mathsf{G}_2'$-invariant discrete metric $d_3: \mathsf{Ein}^{2,3} \times \mathsf{Ein}^{2,3} \rightarrow \{0,1,2,3\}$ on $\mathsf{Ein}^{2,3}$. In fact, we show $\mathsf{G}_2' = \mathsf{Isom}(d_3) \cap \mathsf{Diff}(\mathsf{Ein}^{2,3})$. The asymptotic boundary defines a map $α: \mathsf{MS}_{k} \rightarrow \mathsf{MP}_{k+6}$ between the equidimensional moduli spaces of holomorphic polynomial sextic differentials of degree $k$ and of annihilator polygons with $k+6$ vertices and is conjectured to be a homeomorphism onto its image. We also discuss the relationship between $ν_q$ and a related minimal surface $f_q: \mathbb{C} \rightarrow \mathsf{G}_2'/K$ in the symmetric space $\mathsf{G}_2'/K$, showing how to realize their mutual harmonic lift to $\mathsf{G}_2'/T$ geometrically. Before beginning the geometry, we prove the existence and uniqueness of a complete (real) solution to the $\mathfrak{g}_2$ affine Toda field equations in $\mathbb{C}$ associated to polynomial $q \in H^0(\mathcal{K}_\mathbb{C}^6)$.

math.DG↗

arXiv2020

Combinatorial Properties of Self-Overlapping Curves and Interior Boundaries

We study the interplay between the recently defined concept of minimum homotopy area and the classical topic of self-overlapping curves. The latter are plane curves which are the image of the boundary of an immersed disk. Our first contribution is to prove new sufficient combinatorial conditions for a curve to be self-overlapping. We show that a curve $γ$ with Whitney index 1 and without any self-overlapping subcurves is self-overlapping. As a corollary, we obtain sufficient conditions for self-overlappingness solely in terms of the Whitney index of the curve and its subcurves. These results follow from our second contribution, which shows that any plane curve $γ$, modulo a basepoint condition, is transformed into an interior boundary by wrapping around $γ$ with Jordan curves. Equivalently, the minimum homotopy area of $γ$ is reduced to the minimal possible threshold, namely the winding area, through wrapping. In fact, we show that $n+1$ wraps suffice, where $γ$ has $n$ vertices. Our third contribution is to prove the equivalence of various definitions of self-overlapping curves and interior boundaries, often implicit in the literature. We also introduce and characterize zero-obstinance curves, further generalizations of interior boundaries defined by optimality in minimum homotopy area.

cs.CG↗