arXiv subjects

Dashen Yan

Publications and source records attributed to Dashen Yan.

4 recordsLinked to original sources

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 nondegenerate $\mathbb{Z}_{2}$-harmonic $1$-forms with shrinking branching sets

We develop a gluing theorem for non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on compact manifolds, in which non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on $\mathbb{R}^{n}$ are glued to the regular zeros of a non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-form. As an immediate consequence, viewing an ordinary harmonic $1$-form as a $\mathbb{Z}_{2}$-harmonic $1$-form without branching set, we prove that for every compact oriented manifold $M^{n}, n\geq 3$, if the first Betti number $b^{1}(M)>0$, then $M$ admits a family of non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms, which resolves a folklore conjecture. We will also discuss several possible applications to special holonomy, in particular, to the field of $G_{2}$-geometry.

math.DG↗

arXiv2025

A Construction of non-degenerate $\mathbb{Z}_{2}$-harmonic functions on $\mathbb{R}^{n}$

We discover an explicit construction of non-degenerate $\mathbb{Z}_{2}$-harmonic functions on $\mathbb{R}^{n},n\geq 3$, using a variant of ellipsoidal coordinates on $\mathbb{R}^{n}$. The branching set of these examples is a codimension-$2$ ellipsoid, providing the first known family of non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on $\mathbb{R}^{n}$ with compact branching sets. Moreover, the graph of the related $\mathbb{Z}_{2}$-harmonic one form in $T^{*}\mathbb{R}^{n}$ can be obtained as a certain limit of a specific sequence of Lawlor's necks in $\mathbb{C}^{n}=T^{*}\mathbb{R}^{n}$.

math.DG↗

arXiv2024

A Gluing Theorem For Collapsing Warped-QAC Calabi-Yau Manifolds

We carry out a gluing construction for collapsing warped-QAC (quasi-asymptotically-conical) Calabi-Yau manifolds in $\CC^{n+2}, n\geq 2$. This gluing theorem verifies a conjecture by Yang Li in \cite{li2019gluing} on the behavior of the warped QAC Calabi-Yau metrics on affine quadrics when two singular fibers of a holomorphic fibration go apart. We will also discuss a bubble tree structure for those collapsing warped-QAC Calabi-Yau manifolds.

math.DG↗