arXiv subjects

Nicholas Rungi

Publications and source records attributed to Nicholas Rungi.

8 recordsLinked to original sources

arXiv2025

Equiaffine immersions and pseudo-Riemannian space forms

We introduce an explicit construction that produces immersions into the pseudosphere $\mathbb{S}^{n,n+1}$ and the pseudohyperbolic space $\mathbb{H}^{n+1,n}$ starting from equiaffine immersions in $\mathbb{R}^{n+1}$, and conversely. We describe how these immersions interact with a para-Sasaki metric defined on $\mathbb{H}^{n+1,n}$ via a principal $\mathbb{R}$-bundle structure over a para-K\"ahler manifold $\mathbb H_\tau^n$, called the para-complex hyperbolic space. In the case where the immersion in $\mathbb{R}^{n+1}$ is an $n$-dimensional hyperbolic affine sphere, we obtain spacelike maximal immersions in $\mathbb{H}^{n+1,n}$ that satisfy a transversality condition with respect to the principal $\mathbb{R}$-bundle structure. As a first application, given a strictly convex subset $\Omega \subset \mathbb{RP}^n$, we define a boundary set $\overline{\Lambda}_\Omega$ in the partial flag variety of lines and hyperplanes in $\mathbb{R}^{n+1}$, and prove the existence and uniqueness of a spacelike, Lagrangian, maximal $n$-submanifold in $\mathbb H^n_\tau$ with boundary $\overline{\Lambda}_\Omega$. We also discuss its implications in the case of $\mathbb H^{n+1,n}$. As a second application, we show that the Blaschke lift of the hyperbolic affine sphere, introduced by Labourie for $n=2$, into the symmetric space of $\mathrm{SL}(n+1,\mathbb{R})$ is a harmonic map.

math.DG↗

arXiv2025

Para-complex geometry and cyclic Higgs bundles

We introduce para-complex and pseudo-Riemannian geometric methods for the study of representations of surface groups in $\mathrm{SL}(2m+1,\mathbb{R})$. For $m=1$ our techniques allow to recover several known results for Hitchin representations without any reference to convex projective geometry or hyperbolic affine spheres. In particular, we describe analytically the Guichard-Wienhard domain of discontinuity in the flag variety and the corresponding concave foliated flag structure of Nolte-Riestenberg. In higher rank, we obtain a one-to-one correspondence between stable cyclic $\mathrm{SL}(2m+1,\mathbb{R})$-Higgs bundles (not necessarily in the Hitchin component) and a special class of surfaces, which we call isotropic $\mathbf{P}$-alternating, in the para-complex hyperbolic space $\mathbb{H}^{2m}_{\tau}$. As a result, we give a geometric interpretation to the holomorphic differential $q_{2m+1}$ in the Hitchin base in terms of harmonic sequences for immersions in para-complex manifolds.

math.DG↗

arXiv2024

Complex Lagrangian minimal surfaces, bi-complex Higgs bundles and $\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations

In this paper we introduce complex minimal Lagrangian surfaces in the bi-complex hyperbolic space and study their relation with representations in $\mathrm{SL}(3,\mathbb{C})$. Our theory generalizes at the same time minimal Lagrangian surfaces in the complex hyperbolic plane, hyperbolic affine spheres in $\mathbb{R}^3$, and Bers embeddings in the holomorphic space form $\mathbb{CP}^1 \times \mathbb{CP}^1 \setminus \Delta$. If these surfaces are equivariant under representations in $\mathrm{SL}(3,\mathbb{C})$, our approach generalizes the study of almost $\mathbb{R}$-Fuchsian representations in $\mathrm{SU}(2,1)$, Hitchin representations in $\mathrm{SL}(3,\mathbb{R})$, and quasi-Fuchsian representations in $\mathrm{SL}(2,\mathbb{C})$. Moreover, we give a parameterization of $\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations by an open set in the product of two copies of the bundle of holomorphic cubic differentials over the Teichm\"uller space of $S$, from which we deduce that this space of representations is endowed with a bi-complex structure. In the process, we introduce bi-complex Higgs bundles as a new tool for studying representations into semisimple complex Lie groups.

math.DG↗

arXiv2023

The moduli space of flat maximal space-like embeddings in pseudo-hyperbolic space

We study the moduli space of flat maximal space-like embeddings in $\mathbb{H}^{2,2}$ from various aspects. We first describe the associated Codazzi tensors to the embedding in the general setting, and then, we introduce a family of pseudo-K\"ahler metrics on the moduli space. We show the existence of two Hamiltonian actions with associated moment maps and use them to find a geometric global Darboux frame for any symplectic form in the above family.

math.DG↗

arXiv2023

Riemannian geometry of maximal surface group representations acting on pseudo-hyperbolic space

For any maximal surface group representation into $\mathrm{SO}_0(2,n+1)$, we introduce a non-degenerate scalar product on the the first cohomology group of the surface with values in the associated flat bundle. In particular, it gives rise to a non-degenerate Riemannian metric on the smooth locus of the subset consisting of maximal representations inside the character variety. In the case $n=2$, we carefully study the properties of the Riemannian metric on the maximal connected components, proving that it is compatible with the orbifold structure and finding some totally geodesic sub-varieties. Then, in the general case, we explain when a representation with Zariski closure contained in $\mathrm{SO}_0(2,3)$ represents a smooth or orbifold point in the maximal $\mathrm{SO}_0(2,n+1)$-character variety and we show that the associated space is totally geodesic for any $n\ge 3$.

math.DG↗

arXiv2023

Pseudo-K\"ahler structure on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component and Goldman symplectic form

The aim of this paper is to show the existence and give an explicit description of a pseudo-Riemannian metric and a symplectic form on the $\mathrm{S}\mathrm{L}(3,\mathbb{R})$-Hitchin component, both compatible with Labourie and Loftin's complex structure. In particular, they give rise to a mapping class group invariant pseudo-K\"ahler structure on a neighborhood of the Fuchsian locus, which restricts to a multiple of the Weil-Petersson metric on Teichm\"uller space. By comparing our symplectic form with Goldman's $\boldsymbol{\omega}_G$, we prove that the pair $(\boldsymbol{\omega}_G, \mathbf{I})$ cannot define a K\"ahler structure on the Hitchin component.

math.DG↗

arXiv2022

Global Darboux coordinates for complete Lagrangian fibrations and an application to the deformation space of $\mathbb{R}\mathbb{P}^2$-structures in genus one

In this paper we study a broad class of complete Hamiltonian integrable systems, namely the ones whose associated Lagrangian fibration is complete and has non compact fibres. By studying the associated complete Lagrangian fibration, we show that, under suitable assumptions, the integrals of motion can be taken as action coordinates for the Hamiltonian system. As an application we find global Darboux coordinates for a new family of symplectic forms $\boldsymbol{\omega}_f$, parametrized by smooth functions $f:[0,+\infty)\to(-\infty,0]$, defined on the deformation space of properly convex $\mathbb{R}\mathbb{P}^2$-structures on the torus. Such a symplectic form is part of a family of pseudo-K\"ahler metrics $(\mathbf{g}_f,\mathbf{I},\boldsymbol{\omega}_f)$ defined on $\mathcal{B}_0(T^2)$ and introduced by the authors. In the last part of the paper, by choosing $f(t)=-kt, k>0$ we deduce the expression for an arbitrary isometry of the space.

math.SG↗

arXiv2021

Pseudo-K\"ahler geometry of properly convex projective structures on the torus

In this paper we prove the existence of a pseudo-K\"ahler structure on the deformation space $\mathcal{B}_0(T^2)$ of properly convex $\mathbb R\mathbb P^2$-structures over the torus. In particular, the pseudo-Riemannian metric and the symplectic form are compatible with the complex structure inherited from the identification of $\mathcal{B}_0(T^2)$ with the complement of the zero section of the total space of the bundle of cubic holomorphic differentials over the Teichm\"uller space. We show that the $S^1$-action on $\mathcal{B}_0(T^2)$, given by rotation of the fibers, is Hamiltonian and it preserves both the metric and the symplectic form. Finally, we prove the existence of a moment map for the $\mathrm{SL}(2,\mathbb R)$-action over $\mathcal{B}_0(T^2)$.

math.DG↗