arXiv subjects

Jiahuang Chen

Publications and source records attributed to Jiahuang Chen.

5 recordsLinked to original sources

arXiv2026

Prescribed Singular Sets for Z/2-Harmonic 1-Forms on R^n

Z/2 harmonic 1-forms arise naturally as singular limits in gauge theory and calibrated geometry, but the topology that can occur in their singular sets is poorly understood. In this paper we study the flexibility of these singular sets when the ambient Riemannian metric is allowed to vary. We prove that every compact smoothly embedded codimension-two submanifold with trivial normal bundle in a Euclidean space of dimension at least three can be realized as the singular set of a nondegenerate Z/2 harmonic 1-form for a complete metric that is Euclidean outside a compact set. As an application, we use Calabi surgery to construct Z/2-harmonic 1-forms with prescribed local singular sets on closed manifolds of positive first Betti number.

math.DG↗

arXiv2026

Calabi surgery for Z/2 harmonic 1-forms

We prove a 2-valued analogue of Calabi's intrinsic harmonicity theorem and use it to introduce the Calabi surgery method, a surgery theory for $\mathbb{Z}/2$ harmonic $1$-forms. Once the ambient metric is allowed to vary, the construction of new $\mathbb{Z}/2$ harmonic forms can be reduced to cutting and pasting closed 2-valued 1-forms, matching local harmonic models, and controlling the transitivity of the resulting singular foliation. For these constructions, the Nash--Moser-type analytic deformation problem that arises in singular gluing is replaced by local model matching and a global dynamical condition on the foliation. The resulting procedure gives a flexible way to construct and modify $\mathbb{Z}/2$ harmonic 1-forms under weak regularity assumptions. As applications, we obtain connected-sum and local replacement theorems, blow up isolated ordinary zeros by prescribed Euclidean models, split smooth $\vec{k}$-nondegenerate branching components, and desingularize graphic singular sets in dimensions 3 and 4 with suitable resolution models.

math.DG↗

arXiv2025

On the non-Archimedean Hitchin map for $\mathrm{SL}_2(F)$

Let $F$ be a non-Archimedean valued field, $Σ$ a closed Riemann surface of genus at least two, and $Γ$ its fundamental group. Building on the theory of equivariant harmonic maps into $\mathbb{R}$-trees, we study the non-Archimedean Hitchin map from the $\mathrm{SL}_2(F)$-character variety $\mathcal{X}_F(Γ)$, equipped with the non-Archimedean topology, to the space of holomorphic quadratic differentials on $Σ$. We prove that this map is continuous and that its image is contained in the space of Jenkins--Strebel differentials. Moreover, we establish a dynamical characterization of unbounded representations, showing that the induced action of $Γ$ on the Bruhat--Tits tree of $\mathrm{SL}_2(F)$ is never small.

math.DG↗

arXiv2025

Perturbation and Pruning of Nondegenerate $\mathbb{Z}/2$ Harmonic 1-forms

We prove that for any nondegenerate $\mathbb{Z}/2$ harmonic $1$-form, there exists a metric perturbation producing a new nondegenerate $\mathbb{Z}/2$ harmonic $1$-form whose ordinary zero set is discrete. As an application, we show that for generic smooth nondegenerate $\mathbb{Z}/2$ harmonic $1$-forms, the leaf spaces are $\mathbb{Z}$-trees. Moreover, we show that if a $3$-dimensional rational homology sphere admits a smooth nondegenerate $\mathbb{Z}/2$-harmonic $1$-form, then there exists another nondegenerate $\mathbb{Z}/2$-harmonic $1$-form whose singular locus has exactly two connected components.

math.DG↗

arXiv2024

On the existence and rigidity of critical Z2 eigenvalues

In this article, we study the eigenvalues and eigenfunction problems for the Laplace operator on multivalued functions, defined on the complement of the 2n points on the round sphere. These eigenvalues and eigensections could also be viewed as functions on the configuration spaces of points, introduced and systematically studied by Taubes-Wu. Critical eigenfunctions, which serve as local singularity models for gauge theoretical problems, are of particular interest. Our study focuses on the existence and rigidity problems pertaining to these critical eigenfunctions. We prove that for generic configurations, the critical eigenfunctions do not exist. Furthermore, for each n>1, we construct infinitely many configurations that admit critical eigensections. Additionally, we show that the Taubes-Wu tetrahedral eigensections are deformation rigid and non-degenerate. Our main tools are algebraic identities developed by Taubes-Wu and finite group representation theory.

math.DG↗