arXiv subjects

Charles Ouyang

Publications and source records attributed to Charles Ouyang.

7 recordsLinked to original sources

arXiv2026

Embedded special Legendrian surfaces in $\mathbb S^5$

We construct the first smooth embedded compact special Legendrian surfaces in $\mathbb S^5$ of genus greater than one. More precisely, for every sufficiently large integer $k$, we construct an embedded special Legendrian surface whose conformal structure is the Fermat curve of degree $k$ and genus $\tfrac12(k-1)(k-2)$. Our approach combines an elementary implicit function theorem with the description of special Legendrian surfaces via loop algebra-valued meromorphic connections and a characterization of the unitarizability locus in the ${SL}_{3}(\mathbb C)$-character variety of the thrice-punctured sphere.

math.DG↗

arXiv2023

Higgs bundles and SYZ geometry

Using non-Abelian Hodge theory for parabolic Higgs bundles, we construct infinitely many non-congruent hyperbolic affine spheres modeled on a thrice-punctured sphere with monodromy in $\mathrm{SL}_3(\mathbb{Z})$. These give rise to non-isometric semi-flat Calabi-Yau metrics on special Lagrangian torus bundles over an open ball in $\mathbb{R}^{3}$ minus a Y-vertex, thereby answering a question raised by Loftin, Yau, and Zaslow in [LYZ], [LYZerr].

math.DG↗

arXiv2021

A closed ball compactification of a maximal component via cores of trees

We show that, in the character variety of surface group representations into the Lie group $\mathrm{PSL}(2,\mathbb{R}) \times \mathrm{PSL}(2,\mathbb{R})$, the compactification of the maximal component introduced by the second author is a closed ball upon which the mapping class group acts. We study the dynamics of this action. Finally, we describe the boundary points geometrically as $(\overline{A_{1} \times A_{1}},2)$-valued mixed structures.

math.GT↗

arXiv2021

Boundary of the Gothen components

In this short note we describe an interesting new phenomenon about the $\mathrm{Sp}(4,\mathbb{R})$-character variety. Precisely, we show that the Hitchin component and all Gothen components share the same boundary in our length spectrum compactification.

math.DG↗

arXiv2020

Length spectrum compactification of the $\mathrm{SO}_{0}(2,3)$-Hitchin component

We find a compactification of the $\mathrm{SO}_{0}(2,3)$-Hitchin component by studying the degeneration of the induced metric on the unique equivariant maximal surface in the 4-dimensional pseudo-hyperbolic space $\mathbb{H}^{2,2}$. In the process, we establish the closure in the space of projectivized geodesic currents of the space of flat metrics induced by holomorphic quartic differentials on a Riemann surface. As an application, we describe the behavior of the entropy of Hitchin representations along rays of quartic differentials.

math.DG↗

arXiv2020

Limits of Blaschke metrics

We find a compactification of the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component by studying the degeneration of the Blaschke metrics on the associated equivariant affine spheres. In the process, we establish the closure in the space of projectivized geodesic currents of the space of flat metrics induced by holomorphic cubic differentials on a Riemann surface.

math.DG↗

arXiv2019

High energy harmonic maps and degeneration of minimal surfaces

Let $S$ be a closed surface of genus $g \geq 2$ and let $ρ$ be a maximal $\mathrm{PSL}(2, \mathbb{R}) \times \mathrm{PSL}(2, \mathbb{R})$ surface group representation. By a result of Schoen, there is a unique $ρ$-equivariant minimal surface $\widetildeΣ$ in $\mathbb{H}^{2} \times \mathbb{H}^{2}$. We study the induced metrics on these minimal surfaces and prove the limits are precisely mixed structures. In the second half of the paper, we provide a geometric interpretation: the minimal surfaces $\widetildeΣ$ degenerate to the core of a product of two $\mathbb{R}$-trees. As a consequence, we obtain a compactification of the space of maximal representations of $π_{1}(S)$ into $\mathrm{PSL}(2, \mathbb{R}) \times \mathrm{PSL}(2, \mathbb{R})$.

math.DG↗