On the Moduli Space of Singular Euclidean Surfaces

具有SU(3)结构的6维流形上锥形奇点瞬子的模空间

The moduli space of conically singular instantons over an SU(3)-manifold

本文研究了具有指定切连通性条件的锥形奇点瞬子模空间，发展了针对SU(3)瞬子的Fredholm变形理论，给出了模空间的Kuranishi结构，并推导了瞬子变形算子余核的维数公式。

具正标量曲率的局部共形平坦流形：克莱因群、模空间与欧氏刚性

Locally Conformally Flat Manifolds with Positive Scalar Curvature: Kleinian Groups, Moduli Spaces, and Euclidean Rigidity

该研究探讨具正标量曲率的局部共形平坦流形，通过克莱因群等工具得到宏观维数、同伦群等约束，证明三维相关空间可缩性，构造了不满足欧氏刚性的反例。

$d$-minimal surfaces in three-dimensional singular semi-Euclidean space $\mathbb{R}^{0,2,1}$

In this paper, we investigate surfaces in singular semi-Euclidean space $\mathbb{R}^{0,2,1}$ endowed with a degenerate metric. We define $d$-minimal surfaces, and give a representation formula of Weierstrass type. Moreover, we prove that $d$-minimal surfaces in $\mathbb{R}^{0,2,1}$ and spacelike flat zero mean curvature (ZMC) surfaces in four-dimensional Minkowski space $\mathbb{R}^{4}_{1}$ are in one-to-one correspondence.

到欧氏建筑的调和映射奇异层的可求长性

Rectifiability of the singular strata for harmonic maps to Euclidean buildings

本文为映入$F$-连通复形的调和映射引入奇异层概念，证明其可求长性，并依据[NV17]的可求长Reifenberg纲领给出定量层的Minkowski内容界。

具有分形奇点集的面积极小曲面：普遍性、模空间与分层上的精化

Area-minimizing surfaces with fractal singular sets: prevalence, moduli space, and refinements on strata

本文延续 Almgren 猜想研究，证明了具有分形奇点集的面积极小曲面在一般流形上的普遍性，部分确定了其模空间，并精化了分层结构。

Complete flat fronts as hypersurfaces in Euclidean space

By Hartman--Nirenberg's theorem, any complete flat hypersurface in Euclidean space must be a cylinder over a plane curve. However, if we admit some singularities, there are many non-trivial examples. Flat fronts are flat hypersurfaces with admissible singularities. Murata--Umehara gave a representation formula for complete flat fronts with non-empty singular set in Euclidean $3$-space, and proved the four vertex type theorem. In this paper, we prove that, unlike the case of $n=2$, there do not exist any complete flat fronts with non-empty singular set in Euclidean $(n+1)$-space $(n\geq 3)$.

Moduli space and deformations of special Lagrangian submanifolds with edge singularities

Special Lagrangian submanifolds are submanifolds of a Calabi-Yau manifold calibrated by the real part of the holomorphic volume form. In this paper we use elliptic theory for edge-degenerate differential operators on singular manifolds to study the moduli space of deformations of special Lagrangian submanifolds with edge singularities. We obtain a general theorem describing the local structure of the moduli space. When the obstruction space vanishes the moduli space is a smooth, finite dimensional manifold.

SYZ Mirror Symmetry of Hitchin's Moduli Spaces Near Singular Fibers I

We study hyperkahler metrics and hyperholomorphic connections of Hitchin's moduli spaces after Gaiotto, Moore and Neitzke. Their construction via the twistor technique produces intricate wall crossing behaviors. For certain four dimensional Hitchin's moduli spaces local models and degeneration to local models near singular fibers of the Hitchin's fibration are understood.

平坦曲面的模空间体积

Volumes of moduli spaces of flat surfaces

本文研究了具有预定锥奇点的平坦曲面模空间，证明了其体积有限，并通过对其穿孔曲面的 Euler 示性数进行归纳，证明了体积在几乎所有奇点阶数下均可显式计算。

A compactification of the moduli space of principal Higgs bundles over singular curves

A principal Higgs bundle $(P,ϕ)$ over a singular curve $X$ is a pair consisting of a principal bundle $P$ and a morphism $ϕ:X\to\text{Ad}P \otimes Ω^1_X$. We construct the moduli space of principal Higgs G-bundles over an irreducible singular curve $X$ using the theory of decorated vector bundles. More precisely, given a faithful representation $ρ:G\to Sl(V)$ of $G$, we consider principal Higgs bundles as triples $(E,q,ϕ)$ where $E$ is a vector bundle with $\rk{E}=\dim V$ over the normalization $\xtilde$ of $X$, $q$ is a parabolic structure on $E$ and $ϕ:E\ab{}\to L$ is a morphism of bundles, …

Local and nonlocal singular Liouville equations in Euclidean spaces

We study metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$(-Δ)^\frac{n}{2}w=e^{nw}-cδ_{0} \text{ on } \mathbb R^n,$$ under a finite volume condition. We analyze the asymptotic behaviour at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.

The symplectic geometry of polygons in hyperbolic 3-space

We study the symplectic geometry of the moduli space of closed n-gons with fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces have a symplectic structure coming from Poisson Lie theory. We construct completely integrable systems on these moduli spaces by bending n-gons along their diagonals. The results of this paper are the analogues of the results of the first two authors for n-gon linkages in Euclidean 3-space in JDG 44. We conclude by proving by a deformation argument that the moduli spaces of n-gon linkages in hyperbolic 3-space and Euclidean 3-space with the same…

Higher dimensional Scherk's hypersurfaces

In 3-dimensional Euclidean space, Scherk second surfaces are singly periodic embedded minimal surfaces with four planar ends. In this paper, we obtain a natural generalization of these minimal surfaces in any higher dimensional Euclidean space ${\R}^{n+1}$, for $n \geq 3$. More precisely, we show that there exist $(n-1)$-periodic embedded minimal hypersurfaces with four hyperplanar ends. The moduli space of these hypersurfaces forms a 1-dimensional fibration over the moduli space of flat tori in ${\R}^{n-1}$. A partial description of the boundary of this moduli space is also given.

Hitchin 模空间中的奇异拉格朗日子流形与共形极限

Singular Lagrangians in the Hitchin moduli space and conformal limits

本文在 SL(r, C)-Higgs 丛模空间中构造了通过多稳定不动点的拉格朗日子轨迹，并在特定条件下证明了其共形极限的存在性。

Moduli Spaces of Singular Yamabe Metrics

Complete, conformally flat metrics of constant positive scalar curvature on the complement of $k$ points in the $n$-sphere, $k \ge 2$, $n \ge 3$, were constructed by R\. Schoen [S2]. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of independent interest. The main result is that the moduli space is a locally rea…

在欧几里得3空间中，线性依赖条件下翻译表面的奇点

Singularities of translation surfaces under the linearly dependent condition in Euclidean 3-space

本文研究了在欧几里得3空间中，线性依赖条件下翻译表面的奇点特性，通过广义框架表面理论分析奇点类型。

中心双曲2-单极子模空间的渐近结构

The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space

本文在点粒子近似下构造了两个中心双曲单极子模空间的渐近度量，通过考虑对径构型克服了双曲情形下无法分解质心运动的困难，得到了负质量Taub-NUT度量的双曲类比，并建立了其与Hitchin构造的双曲Atiyah-Hitchin度量渐近形式的关系。

欧氏空间中前曲线螺旋面的奇点

Singularities of helicoidal surfaces of frontals in the Euclidean space

本文研究欧氏空间中由含奇点前曲线生成的螺旋面，将其刻画为广义标架基曲面，并利用 Legendre 曲线曲率给出其不变量、曲率及奇点判据。

Non-abelian vortices, Hecke modifications and singular monopoles

In this note we show that for the group G = U(N) the space of Hecke modifications of a rank N vector bundle over a Riemann surface C coincides with the moduli space of solutions of certain non-abelian vortex equations over C . Through the recent work of Kapustin and Witten this then leads to an isomorphism between the moduli space of vortices and the moduli space of singular monopoles on the product of C with a closed interval I .

Piecewise Euclidean structures and Eberlein's Rigidity Theorem in the singular case

In this article, we generalize Eberlein's Rigidity Theorem to the singular case, namely, one of the spaces is only assumed to be a CAT(0) topological manifold. As a corollary, we get that any compact irreducible but locally reducible locally symmetric space of noncompact type does not admit a nonpositively curved (in the Aleksandrov sense) piecewise Euclidean structure. Any hyperbolic manifold, on the other hand, does admit such a structure.

A metric on the moduli space of bodies

We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of symplectic toric manifolds. We also discuss related problems.

Best Sobolev constants in the presence of sharp Hardy terms in Euclidean and hyperbolic space

In this article we compute the best Sobolev constants for various Hardy-Sobolev inequalities with sharp Hardy term. This is carried out in three different environments: interior point singularity in Euclidean space, interior point singularity in hyperbolic space and boundary point singularity in Euclidean domains.

Partial Resolutions of Orbifold Singularities via Moduli Spaces of HYM-type Bundles

Let $Γ$ be a finite group acting linearly on $\C^n$, freely outside the origin, and let $N$ be the number of conjugacy classes of $Γ$ minus one. A construction of Kronheimer of moduli spaces $X_ζ$ of translation-invariant $Γ$-equivariant instantons on $\C^2$ is generalised to $\C^n$. The moduli spaces $X_ζ$ depend on a parameter $ζ\in\Q^N$. The following results are proved: for $ζ=0$, $X_0$ is isomorphic to $\C^n/Γ$; if $ζ\neq 0$, the natural maps $X_ζ\to X_0$ are partial resolutions. The moduli $X_ζ$ are furthermore shown to admit Kähler metrics which are Asymptotically Locally Euclidean (ALE…