Hyperbolic monopoles and holomorphic spheres

$L^2$ 几何中的双曲磁单极子

$L^2$ geometry of hyperbolic monopoles

本文针对双曲磁单极子模空间上 $L^2$ 度量发散的问题，提出受超对称启发的规范固定条件加以消除，得到双曲类比于欧几里得情形的超凯勒几何。

具有连续对称性的双曲单极子

Hyperbolic monopoles with continuous symmetries

本文建立连续对称双曲单极子的分类框架与结构定理，将球对称情形归结为表示论问题，并给出结构群约束，进而构造新的球对称 $\mathrm{Sp}(n)$ 双曲单极子。

The Homotopy Type of hyperbolic Monopole Orbit Spaces

We prove that the space of gauge equivalence classes of U(1)-invariant connections on some SU(2)-principle bundles over the 4-sphere S^4 is weakly homotopy equivalent to a component of the second loop space of the 2-sphere S^2.

Boundary algebras of hyperbolic monopoles

We prove the conjecture that a monopole in three-dimensional anti-de Sitter space can be completely determined by its ``holographic'' image on the conformal boundary two-sphere.

双曲球中的共形重心与全纯重心

Conformal and holomorphic barycenters in hyperbolic balls

本文在双曲球中定义可测集的共形重心与全纯重心，指出二者在圆盘重合但在多维球中不同，推广了Douady-Earle球面重心理论。

Homogeneous Kobayashi-hyperbolic manifolds with high-dimensional group of holomorphic automorphisms

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-2$. This result complements an existing classification for automorphism group dimension $n^2-1$ and greater obtained without the homogeneity assumption.

Algebraicity of analytic maps to a hyperbolic variety

Let $X$ be an algebraic variety over $\mathbb{C}$. We say that $X$ is Borel hyperbolic if, for every finite type reduced scheme $S$ over $\mathbb{C}$, every holomorphic map $S^{an}\to X^{an}$ is algebraic. We use a transcendental specialization technique to prove that $X$ is Borel hyperbolic if and only if, for every smooth affine curve $C$ over $\mathbb{C}$, every holomorphic map $C^{an}\to X^{an}$ is algebraic. We use the latter result to prove that Borel hyperbolicity shares many common features with other notions of hyperbolicity such as Kobayashi hyperbolicity.

Hyperbolic Multi-Monopoles With Arbitrary Mass

On a complete manifold, such as Euclidean 3-space or hyperbolic 3-space, the limit at infinity of the norm of the Higgs field is called the mass of the monopole. We show the existence, on hypebolic 3-space, of monopoles with given magnetic charge and arbitrary mass. Previously, aside from charge one monopoles, existence was known only for monopoles with integral mass (since these arise from U(1) invariant instantons on Euclidean 4-space). The method of proof is based on Taubes' gluing procedure, using well-separated, explicit, charge one monopoles. The analysis is carried out in a weighted Sob…

Pluricomplex geometry and hyperbolic monopoles

Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a 2-sphere of complex structures, but they no longer behave like unit imaginary quaternions. We still require, however, that the 2-sphere of complex structures determines a decomposition of the complexified tangent space as tensor product of C^{2n} and C^2. Among interesting properties of pluricomplex manifold is the existence of a canonical torsion free connec…

双曲2单极子的$L^2$度量

The $L^2$ metric for hyperbolic 2-monopoles

本文计算了质量为$\ frac{1}{2}$的2单极子的修正$L^2$度量，该度量为复非退化双线性型，在反演对称2单极子子空间上为黎曼度量，且可通过初等函数与椭圆积分显式算出，其渐近形式与文献预期一致。

关于横向全纯部分双曲流的研究

On transversely holomorphic partially hyperbolic flows

本文研究横向全纯部分双曲流，证明在七维情况下，此类流可投影为横向全纯Anosov流，并在特定条件下与复流形或双曲流形的测地流轨道等价。

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。

Hermitian-holomorphic Deligne cohomology, Deligne pairing for singular metrics, and hyperbolic metrics

We introduce a model for Hermitian holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup-product in cohomology is shown to correspond to a generalization of the Deligne pairing to line bundles with "good" hermitian metrics in the sense of Mumford and others. A particular case is that of the tangent bundle of the curve twisted by the negative of the singularity divisor of a hyperbolic metric: its cup square (corrected by the total area) is shown to b…

Seiberg--Witten Monopoles on Seifert Fibered Spaces

In this paper, we investigate the Seiberg-Witten gauge theory for Seifert fibered spaces. The monopoles over these three-manifolds, for a particular choice of metric and perturbation, are completely described. Gradient flow lines between monopoles are identified with holomorphic data on an associated ruled surface, and a dimension formula for such flows is calculated.

Singular Hermitian-Einstein monopoles on the product of a circle and a Riemann surface

In this paper, the moduli space of singular unitary Hermitian--Einstein monopoles on the product of a circle and a Riemann surface is shown to correspond to a moduli space of stable pairs on the Riemann surface. These pairs consist of a holomorphic vector bundle on the surface and a meromorphic automorphism of the bundle. The singularities of this automorphism correspond to the singularities of the singular monopole. We then consider the complex geometry of the moduli space; in particular, we compute dimensions, both from the complex geometric and the gauge theoretic point of view.

Rigidity of proper holomorphic mappings between certain unbounded non-hyperbolic domains

The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbf{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbf{C}^n\times \mathbf{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbf{C}^{n+m}$. Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock-Bargmann-Hartogs domains in terms of the polylogarithm functions and Kim-Ninh-Yamamori determined the automorphism group of the domain $D_{n,m}(μ)$. In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock-Bargmann-Hartogs …

关于超凯勒簇的反双曲性

On anti-hyperbolicity for hyperkähler varieties

本文研究超凯勒流形被同维数复仿射空间全纯或亚纯支配的反双曲性判据与例子，推广了 K3 曲面的相关已知结果。

Hodge theory of holomorphic vector bundle on compact Kähler hyperbolic manifold

Let $E$ be a holomorphic vector bundle over a compact Kähler manifold $(X,ω)$ with negative sectional curvature $sec\leq -K<0$, $Δ_{E}$ be the Chern connection on $E$. In this article we show that if $C:=|[Λ,iΘ(E)]|\leq c_{n}K$, then $(X,E)$ satisfy a family of Chern number inequalities. The main idea in our proof is study the $L^{2}$ $\bar{\partial}_{\tilde{E}}$-harmonic forms on lifting bundle $\tilde{E}$ over the universal covering space $\tilde{X}$. We also observe that there is a closely relationship between the eigenvalue of the Laplace-Beltrami operator $Δ_{\bar{\partial}_{\tilde{E}}}$ …

Monopole metrics and the orbifold Yamabe problem

We consider the self-dual conformal classes on n#CP^2 discovered by LeBrun. These depend upon a choice of n points in hyperbolic 3-space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem, which we show is not always solvable (in contrast to the case of compact manifolds). In particular, we show that there is no constant scalar curvature orbifold metric in the c…

部分双曲紧复流形

Partially Hyperbolic Compact Complex Manifolds

该研究提出两类含变体的部分双曲性概念，通过高维整全纯映射及适配增长条件刻画特定方向双曲的紧复流形，证明概念间蕴含关系并给出相关充分判据。

全纯2-球与周转的商空间

Quotients of the holomorphic 2-ball and the turnover

本文在带三个锥点的球面圆盘轨道丛上构造了二维复双曲结构族，突破了此前同类结构局部刚性的限制，并给出闭黎曼曲面上平凡及余切圆盘丛的复双曲结构实例。

Berry's connection, Kähler geometry and the Nahm construction of monopoles

We study supersymmetric deformations of N = 4 quantum mechanics with a Kahler target space admitting a holomorphic isometry. We show that the twisted mass deformation generalises to a deformation constructed from matrix-valued functions of the moment map, which obey the Nahm equations. We also explain how N = 4 supersymmetry implies that the Berry connection on the vacuum bundle for this theory satisfies the BPS monopole equations. In the case where the target space is a Riemann sphere, our analysis reduces to the standard Nahm construction of monopoles. This generalises an earlier result by S…

What to expect from $U(n)$ Seiberg-Witten monopoles for $n > 1$

We study generalisations to the structure groups U(n) of the familiar (abelian) Seiberg-Witten monopole equations on a four-manifold $X$ and their moduli spaces. For $n=1$ one obtains the classical monopole equations. For $n > 1$ our results indicate that there should not be any non-trivial gauge-theoretical invariants which are obtained by the scheme `evaluation of cohomology classes on the fundamental cycle of the moduli space'. For, if $b_2^+$ is positive the moduli space should be `cobordant' to the empty space because we can deform the equations so as the moduli space of the deformed equa…

Harmonic maps and constant mean curvature surfaces in $\H^2 \times \R$

We introduce a hyperbolic Gauss map into the Poincare disk for any surface in H^2xR with regular vertical projection, and prove that if the surface has constant mean curvature H=1/2, this hyperbolic Gauss map is harmonic. Conversely, we show that every nowhere holomorphic harmonic map from an open simply connected Riemann surface into the Poincare disk is the hyperbolic Gauss map of a two-parameter family of such surfaces. As an application we obtain that any holomorphic quadratic differential on the surface can be realized as the Abresch-Rosenberg holomorphic differential of some, and generic…

Immersions of surfaces into SL(2,C) and into the space of geodesics of Hyperbolic space

This thesis mainly treats two developments of the classical theory of hypersurfaces inside pseudo-Riemannian space forms. The former - a joint work with Francesco Bonsante - consists in the study of immersions of smooth manifolds into holomorphic Riemannian space forms of constant curvature -1 (including SL(2,C) with a multiple of its Killing form): this leads to a Gauss-Codazzi theorem, it suggests an approach to holomorphic transitioning of immersions into pseudo-Riemannian space forms, a trick to construct holomorphic maps into the PSL(2,C)-character variety, and leads to a restatement of B…

Weil-Petersson geometry for families of hyperbolic conical Riemann Surfaces

We study the Weil-Petersson geometry for holomorphic families of Riemann Surfaces equipped with the unique conical metric of constant curvature -1.

双曲接触辛提升

Hyperbolic contact symplectic lifts

研究全纯接触流形的双曲性，通过Reeb流形构造具有大自同构群的接触双曲流形，并证明其与Kobayashi双曲辛商的关系。

Hyperbolic Twistor Spaces

In contrast to the classical twistor spaces whose fibres are 2-spheres, we introduce twistor spaces over manifolds with almost quaternionic structures of the second kind in the sense of P. Libermann whose fibres are hyperbolic planes. We discuss two natural almost complex structures on such a twistor space and their holomorphic functions.

The Einstein-Weyl Equations, Scattering Maps, and Holomorphic Disks

We show that conformally compact, globally hyperbolic, Lorentzian Einstein-Weyl 3-manifolds are in natural one-to-one correspondence with orientation-reversing diffeomorphisms of the 2-sphere. The proof hinges on a holomorphic-disk analog of Hitchin's mini-twistor correspondence.

The Haydys monopole equation

We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator; these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to three dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on $\mathbb{R}^3$ is a Kähler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifol…

带边界双曲曲面沿调和映射射线的一致退化

Uniform degeneration of hyperbolic surfaces with boundary along harmonic map rays

本文证明带边界双曲曲面沿调和映射射线退化时，其万有覆盖重标度距离一致收敛到垂直测度叶状结构的交数，并在 Gromov-Hausdorff 意义下收敛到对偶 R-树，从而确定其 Thurston 边界极限。

Monopole Floer homology, eigenform multiplicities and the Seifert-Weber dodecahedral space

We show that the Seifert-Weber dodecahedral space $\mathsf{SW}$ is an $L$-space. The proof builds on our work relating Floer homology and spectral geometry of hyperbolic three-manifolds. A direct application of our previous techniques runs into difficulties arising from the computational complexity of the problem. We overcome this by exploiting the large symmetry group and the arithmetic and tetrahedral group structure of $\mathsf{SW}$ to prove that small eigenvalues on coexact $1$-forms must have large multiplicity.