arXiv subjects

Qiongling Li

Publications and source records attributed to Qiongling Li.

22 records · Page 2Linked to original sources

arXiv2017

Minimal surfaces for Hitchin representations

Given a reductive representation $ρ: π_1(S)\rightarrow G$, there exists a $ρ$-equivariant harmonic map $f$ from the universal cover of a fixed Riemann surface $Σ$ to the symmetric space $G/K$ associated to $G$. If the Hopf differential of $f$ vanishes, the harmonic map is then minimal. In this paper, we investigate the properties of immersed minimal surfaces inside symmetric space associated to a subloci of Hitchin component: $q_n$ and $q_{n-1}$ case. First, we show that the pullback metric of the minimal surface dominates a constant multiple of the hyperbolic metric in the same conformal class and has a strong rigidity property. Secondly, we show that the immersed minimal surface is never tangential to any flat inside the symmetric space. As a direct corollary, the pullback metric of the minimal surface is always strictly negatively curved. In the end, we find a fully decoupled system to approximate the coupled Hitchin system.

math.DG↗

arXiv2015

ADS 3-manifolds and Higgs bundles

In this paper we investigate the relationships between closed AdS 3-manifolds and Higgs bundles. We have a new way to construct AdS structures that allows us to see many of their properties explicitly, for example we can recover the very recent formula by Tholozan for the volumes. We also find applications to the theory of minimal immersions into quadrics with their natural pseudo-Riemannian structure: using the geometry of the AdS manifolds we can characterize the representations admitting equivariant minimal immersions of the Poincare disc into the Klein quadric, the Grassmannian Gr(2,4), and understand the geometry of these minimal immersions.

math.DG↗

arXiv2015

Asymptotics of certain families of Higgs bundles in the Hitchin component

Using Hitchin's parameterization of the Hitchin-Teichmüller component of the $SL(n,\mathbb{R})$ representation variety, we study the asymptotics of certain families of representations. In fact, for certain Higgs bundles in the $SL(n,\mathbb{R})$-Hitchin component, we study the asymptotics of the Hermitian metric solving the Higgs bundle equations. This analysis is used to estimate the asymptotics of the corresponding family of flat connections as we scale the differentials by a real parameter. We consider Higgs fields that have only one holomorphic differential $q_n$ of degree $n$ or $q_{n-1}$ of degree $n-1.$ We also study the asymptotics of the associated family of equivariant harmonic maps to the symmetric space $SL(n,\mathbb{R})/SO(n,\mathbb{R})$ and relate it to recent work of Katzarkov, Noll, Pandit and Simpson.

math.DG↗

arXiv2013

Teichmüller Space Is Totally Geodesic In Goldman Space

We construct a new Riemannian metric on Goldman space $\mathcal{B}(S)$, the space of the equivalence classes of convex projective structures on the surface $S$, and then prove the new metric, as well as the metric of Darvishzadeh and Goldman, restricts to be the Weil-Petersson metric on Teichm$\ddot{u}$ller space, embedded as a submanifold of Goldman space $\mathcal{B}(S)$. Moreover, Teichm$\ddot{u}$ller space endowed with the Weil-Petersson metric then is totally geodesic in the Riemannian manifold $\mathcal{B}(S)$.

math.DG↗