关于Quillen度量的奇异性

Mathai-Quillen形式与Lefschetz理论

Mathai-Quillen forms and Lefschetz theory

利用Mathai-Quillen形式给出了闭流形上光滑映射的莱夫谢茨数的一个积分公式.应用于恒等映射，这个公式简化为Chern-Gauss-Bonnet定理。该公式是计算显式常曲率度量。实际上存在一个单参数的整式族。当参数趋于无穷大时，得到了Lefschetz不动点子流形公式的热方程证明的拓扑形式。当参数趋于零并且在横截性假设下，获得映射到其割迹中的点的数量的下界。对于莱夫谢茨数不等于欧拉示性数的自同态，这个数对于大多数度量是无限的，特别是对于非正曲率的度量。

θ丛、Quillen联络与霍奇理论射影结构

Theta bundle, Quillen connection and the Hodge theoretic projective structure

该研究针对亏格≥2紧黎曼曲面模空间上霍奇理论射影结构族，借助霍奇线丛行列式对偶的复联络刻画，构造对应联络并给出三种等价描述，完善射影结构与联络的对应关系。

Hodge丛上度量的奇异性及其拓扑不变量

Singularities of metrics on Hodge bundles and their topological invariants

我们考虑复投射Calabi-Yau簇的退化，并研究了Hodge和行列式丛上的$L^2$，Quillen和BCOV度量的奇异性.在扩张的度量接近非光滑纤维的显性和次显性项被证明是与众所周知的拓扑不变量的奇异性，如极限霍奇结构，消失的周期和对数正则阈值。我们还描述了相应的不变量更一般的退化家庭的情况下，奎伦度量。

具有$S^1$作用的PDKEEPAAA流形上解析挠率的反常公式

The anomaly formula of the analytic torsion on CR manifolds with $S^1$ action

设$X$是一个紧连通强伪凸CR流形，其维数为$2n+1, n\ge 1$，且在$X$上有一个横截CRPDMATHABMO-作用。本文在$X$上的Kohn-Rossi上同调的Fourier分量关于$S^1$作用的行列式线上引入Quillen度量.研究了流形$X$上和$X$上向量丛上的Quillen度量在度量变化时的行为.得到了$X$上Quillen度量关于$S^1$-作用量的反常公式.

Quillen度量变化的渐近展开及其moment map解释

Asymptotic expansion of the variation of the Quillen metric and its moment map interpretation

本文基于equivariant确定线丛框架，将Kähler度量的标量曲率解释为compatible almost complex结构空间上的moment map，同时推导了Quillen度量对数变化的渐近展开。

量子霍尔效应与奎伦度规

Quantum Hall effect and Quillen metric

研究了紧致黎曼曲面上整数量子霍尔效应的生成泛函、绝热曲率和绝热相位。对于生成泛函，我们导出了它在磁场的大通量下的渐近展开式，即，对于正厄米特线丛$L^k$的大次$k$.展开式由反常项和正合项组成。反常项是膨胀的主导项。该部分负责量子化绝热输运系数的QHE。然后，我们确定的Quillen度量的行列式线丛上的非本地（异常）的部分的扩展，和subleading确切的部分与渐近的正则化谱行列式的线丛$L^k$，在大$k$。最后，我们展示了整数QHE的生成泛函与规范和引力（2+1）d Chern-Simons泛函的关系。我们观察到Quillen度规的Bismut-Gillet-Soulé曲率公式与整数量子Hall态的电磁和几何绝热输运的绝热曲率之间的关系。然后，我们在QHE的绝热相的eta不变量，并表明，绝热相的几何部分给出的Chern-Simons功能。

深度1轨形奇点解消上的常标量曲率Kähler度量

Constant Scalar Curvature Kähler Metrics on Resolutions of an Orbifold Singularity of Depth 1

本文针对沿高复余维子轨形具有I型奇点的常标量曲率Kähler轨形，构造其适当解消上的常标量曲率Kähler度量新例。

Kahler-Einstein metric near an isolated log canonical singularity

We construct Kahler-Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2 or if the singularity is smoothable. In complex dimension 2, we show that any complete Kahler-Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to one of the model metrics constructed by Kobayashi and Nakamura arising from hyperbolic geometry.

On the construction of a complete Kahler-Einstein metric with negative scalar curvature near an isolated log-canonical singularity

In this short note we are concerned with the Kahler-Einstein metrics near cone type log canonical singularities. By two different approaches, we construct a complete Kahler-Einstein metric with negative scalar curvature in a neighborhood of the cone over a Calabi-Yau manifold, which provides a local model for the future study of the global Kahler-Einstein metrics on singular varieties. In the first approach, we show that the singularity is uniformized by a complex ball and hence the induced metric from the Bergman metric of the ball is a desired one. In the second approach, we obtain a complet…

Isolated singularities of flat metrics on Riemann surfaces

Robert Bryant (Theorie des varietes minimales et applications, 1988, 154: 321-347) proved that an isolated singularity of a conformal metric of positive constant curvature on a Riemann surface is a conical one. Using Complex Analysis, we find all of the local models for an isolated singularity of a flat metric whose area satisfies some polynomial growth condition near the singularity. In particular, we show that an isolated singularity of a flat metric with finite area is also a conical one.

孤立对数终端奇点附近具有正曲率的Kähler-Einstein度量

Kähler-Einstein metrics with positive curvature near an isolated log terminal singularity

研究孤立对数终端奇点附近正曲率Kähler-Einstein度量的存在性，将其转化为复Monge-Ampère方程的Dirichlet问题，证明次临界区域下Moser-Trudinger不等式成立及光滑解存在，且临界指数可用奇点的归一化体积这一代数不变量表示。

Ricci Flow Singularity for Triaxial Bianchi IX Metric

In this work, we are going to find sufficient conditions on the initial triaxial Bianchi IX metric on some 4-dimensional manifolds foliated by homogeneous S3 for a Type I singularity to occur when it is flowed under the Ricci flow. This work generalises the study on rotationally symmetric manifolds done by Angenent and Isenberg as well as the work of Isenberg, Knopf, and Sesum, in which they introduced some ansatz for the problem setup.

G $_2$流形孤立奇点附近的渐近收敛速度

Rate of asymptotic convergence near isolated singularity of G$_2$ manifold

本文在旗流形上锥内球上构造了一个具有G $_2$完整性和慢收敛速率的度量.

Singularity theorems for $C^1$-Lorentzian metrics

Continuing recent efforts in extending the classical singularity theorems of General Relativity to low regularity metrics, we give a complete proof of both the Hawking and the Penrose singularity theorem for $C^1$-Lorentzian metrics - a regularity where one still has existence but not uniqueness for solutions of the geodesic equation. The proofs make use of careful estimates of the curvature of approximating smooth metrics and certain stability properties of long existence times for causal geodesics. On the way we also prove that for globally hyperbolic spacetimes with a $C^1$-metric causal ge…

正数量曲率与孤立锥形奇点

Positive scalar curvature and isolated conical singularity

本文证明孤立锥形奇点的Geroch型结果，指出$X \\# T^n$不存在带孤立锥形奇点且正则部分非负、某点正数量曲率的度量，还证明具有限孤立锥形奇点的数量平坦度量必平坦且可光滑延拓过奇点。

$C^{1,1}$ -度量的Hawking奇异性定理

Hawking's singularity theorem for $C^{1,1}$-metrics

我们在正则类$C^{1,1}$中给出了霍金奇异性定理的详细证明，即，对于具有局部Lipschitz连续一阶导数的时空度量。证明使用最近的结果在$C^{1,1}$ -因果关系理论，是基于正则化技术适应的因果结构。

Analytic torsion for surfaces with cusps I. Compact perturbation theorem and anomaly formula

Let $\overline{M}$ be a compact Riemann surface and let $g^{TM}$ be a metric over $\overline{M} \setminus D_M$, where $D_M \subset \overline{M}$ is a finite set of points. We suppose that $g^{TM}$ is equal to the Poincaré metric over a punctured disks around the points of $D_M$. The metric $g^{TM}$ endows the twisted canonical line bundle $ω_M(D)$ with the induced Hermitian norm $\|\cdot\|_M$ over $\overline{M} \setminus D_M$. Let $(ξ, h^ξ)$ be a holomorphic Hermitian vector bundle over $\overline{M}$. In this article we define the analytic torsion $T(g^{TM}, h^ξ \otimes \|\cdot\|_M^{2n})$ ass…

奇点类型的度量几何

The metric geometry of singularity types

本文在紧Kähler流形上的奇点类型空间中引入自然伪度量，证明其在正质量下完备，并应用于复Monge-Ampère方程解的收敛性与乘子理想层的半连续性。

Isolated singularities of conformal hyperbolic metrics

J. Nitsche proved that an isolated singularity of a conformal hyperbolic metric is either a conical singularity or a cusp one. We prove by developing map that there exists a complex coordinate $z$ centered at the singularity where the metric has the expression of either $\displaystyle{\frac{4α^2\vert z \vert^{2α-2}}{(1-\vert z \vert ^{2α})^2}\vert \mathrm{d} z \vert^2}$ with $α>0$ or $\displaystyle{\vert z \vert ^{-2}\big(\ln|z|\big)^{-2}|dz|^{2}}$.

Constant $Q$-curvature metrics with a singularity

For dimensions $n \geq 3$, we classify singular solutions to the generalized Liouville equation $(-Δ)^{n/2} u = e^{nu}$ on $\mathbb{R}^n \setminus \{0\}$ with the finite integral condition $\int_{\mathbb{R}^n} e^{nu} < \infty$ in terms of their behavior at $0$ and $\infty$. These solutions correspond to metrics of constant $Q$-curvature which are singular in the origin. Conversely, we give an optimal existence result for radial solutions. This extends some recent results on solutions with singularities of logarithmic type to allow for singularities of arbitrary order. As a key tool to the exis…

数值有效Kähler-Ricci流的奇点类型的独立性

Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows

本文证明了在数值有效流形上，Kähler-Ricci流解的奇点类型不依赖于初始度量的选择，推广了Y. Zhang在半丰富情形下的结果。

霍金奇点定理对于利普希茨洛伦兹度量

Hawking's singularity theorem for Lipschitz Lorentzian metrics

该研究证明了霍金奇点定理在局部利普希茨正则性度量下的适用性，通过新估计和世界体积估计替代传统聚焦技术。

Removable singularity of positive mass theorem with continuous metrics

In this paper, we consider asymptotically flat Riemannnian manifolds $(M^n,g)$ with $C^0$ metric $g$ and $g$ is smooth away from a closed bounded subset $Σ$ and the scalar curvature $R_g\ge 0$ on $M\setminus Σ$. For given $n\le p\le \infty$, if $g\in C^0\cap W^{1,p}$ and the Hausdorff measure $\mathcal{H}^{n-\frac{p}{p-1}}(Σ)<\infty$ when $n\le p<\infty$ or $\mathcal{H}^{n-1}(Σ)=0$ when $p=\infty$, then we prove that the ADM mass of each end is nonnegative. Furthermore, if the ADM mass of some end is zero, then we prove that $(M^n,g)$ is isometric to the Euclidean space by showing the manifold…

Kähler-里奇流的无限时间奇点型

Infinite-time singularity type of the Kähler-Ricci flow

对于具有半充分标准线丛的紧致Kähler流形上的Kähler-里奇流，我们证明了无穷远点的奇点类型不依赖于初始度量的选择.对已有的Kähler-里奇流无限时间奇异型的分类结果，我们也给出了新的简单证明.

Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow

For any manifold $N^p$ admitting an Einstein metric with positive Einstein constant, we study the behavior of the Ricci flow on high-dimensional products $M = N^p \times S^{q+1}$ with doubly-warped product metrics. In particular, we provide a rigorous construction of local, type II, conical singularity formation on such spaces. It is shown that for any $k > 1$ there exists a solution with curvature blow-up rate $\| Rm \|_{\infty} (t) \gtrsim (T-t)^{-k}$ with singularity modeled on a Ricci-flat cone at parabolic scales.

霍金奇点定理的Hölder连续度量的低正则性版本

The Hawking Singularity Theorem for Hölder Continuous Metrics with $L^p$-Bounded Curvature

本文在低正则性条件下证明了霍金奇点定理，研究了时空度量的正则性和曲率限制，扩展了之前的结果并证明了因果平坦性和Myers定理的低正则性版本。

Hawking-Penrose 奇点定理在 $C^{1,1}$-Lorentzian 度量下的成立性

The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics

本文证明 Hawking-Penrose 奇点定理及其推广在 $C^{1,1}$-Lorentzian 度量下依然成立，通过提出弱版本的能量条件与一般性条件，并利用正则化方法分析矩阵 Riccati 方程，证明因果测地线必然变为非最大化的。

正则性$C^{1,1}$中的Penrose奇点定理

The Penrose singularity theorem in regularity $C^{1,1}$

我们将Penrose奇异性定理的有效性推广到正则性时空度量$C^{1,1}$。证明是基于正则化技术，结合最近的结果，在低规律性因果关系理论。

具有上曲率界的空间的墙奇异性

Wall singularity of spaces with an upper curvature bound

本文研究了具有上曲率界的空间中一维余维的墙奇异性，提出了余维一奇异性的几何结构定理和余维二正则性的几何刻画，给出了奇异集余维至少为二的充要条件。

解析Reissner-Nordstrom奇点

Analytic Reissner-Nordstrom Singularity

的Reissner-Nordstrom解决方案和超越奇点的解析扩展。通过使用新的坐标获得的扩展，在该坐标中，度量在$r=0$处退化。度量在新的坐标系中仍然是奇异的，但它的分量变得有限和光滑。利用这一推广，它表明，充电和非旋转的黑洞奇点是兼容的整体双曲性和与守恒的初始值数据。得到了带电粒子的几何模型。

负锥奇异闭Riemann surface上的高斯曲率

Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case

研究了具有圆锥奇点的黎曼曲面的高斯曲率的确定问题。设$(Σ,β)$是一个具有因子$β$和$K_λ=K+λ$的闭黎曼曲面，其中$K:Σ\rightarrow\mathbb{R}$是一个满足$\max_ΣK= 0$，$K\not\equiv 0$和$λ\in\mathbb{R}$的Hölder连续函数。若欧拉特征$χ(Σ,β)$为负，则用变分方法证明了存在一个常数$λ^\ast>0$，使得对任意$λ\leq 0$，存在唯一的共形度量，其高斯曲率为$K_λ$，对任意$λ$，$0<λ<λ^\ast$，至少存在两个共形度量，其高斯曲率为$K_λ$;对于$λ=λ^\ast$，存在至少一个具有高斯曲率$K_{λ^\ast}$的共形度量;对于任何$λ>λ^\ast$，不存在具有其高斯曲率的$K_λ$的特定共形度量。这个结果与Ding和Liu的\cite{Ding-Liu}的结果类似，部分类似于Borer、Galimberti和Struwe的\cite{B-G-Stru}的结果，并且推广了Troyanov的\cite{Troyanov}的负情况。

偶锥球面度量：锥奇点处的爆破

Even Cone Spherical Metrics: Blow-Up at a Cone Singularity

本文研究平坦环面上具有指定锥奇点爆破行为的偶球面度量族，证明其存在性由环面几何决定，并给出与单锥奇点族的显式变换及菱形环面的刚性结果。

拟投射流形上的里奇流

Ricci flow on quasiprojective manifolds

我们考虑紧致Kaehler流形X中完全有限体积度量上的Kaehler-里奇流，度量存在于除数的补上。假设在初始度量上有一定的空间渐近性，我们根据X上的上同调数据计算奇异时间。我们也给出了一个充分条件的奇点，如果有一个，是II型。

黎曼面的退化与拉普拉斯算子的小特征值

Degeneration of Riemann surfaces and small eigenvalues of the Laplacian

本文研究了黎曼面退化过程中拉普拉斯算子小特征值的行为，通过热核估计和奎林度量两种方法分析了其渐进行为。

矩形环面上带一个锥形奇点的球度量的精确计数

Exact counting of spherical metrics with one conical singularity on rectangular tori

该研究针对矩形环面上带一个锥形奇点的球度量，确定了奇异 Liouville 方程的解的计数，通过结合已有结果得出不同参数下球度量的存在性与数量，并发展了适用于全参数范围的统一方法。

关于Poincaré-Einstein度规的锥退化和弯曲的几点注记

Some remarks on conic degeneration and bending of Poincaré-Einstein metrics

设$(M,g)$是具有$c_1 > 0$的紧致Kähler-Einstein流形.用$K\to M$表示正则线丛，其全空间为$X$，$X_0$表示通过沿其零截面吹落$X$而获得的奇异空间。我们采用了一个建设的页面和教皇，并讨论了一个有趣的多参数家庭的庞加莱-爱因斯坦度量$X$。其中一个单参数子集$\{g_t\}_{t>0}$具有如下性质：与$t\searrow 0$一样，$g_t$收敛于$X_0$上的PE度量$g_0$，而$t^{-1}g_t$收敛于$X$上的完全Ricci平坦Kähler度量$\hat{g}_0$.另一个1参数的子族有一个边缘奇点沿$X$的零部分，锥角取决于参数，但有恒定的共形无穷大。这说明了庞加莱-爱因斯坦模空间的一些意想不到的特征。

奇点几何与黑洞信息悖论

The geometry of singularities and the black hole information paradox

蒸发黑洞只有在时间演化终止于奇点时才会发生信息损失。但是，正如我们将要看到的，黑洞解允许在奇点之外的解析扩展，到整体双曲解。所用的方法类似于在事件视界的视奇异性，但在奇异性处，所得到的度量是退化的。当度规退化时，协变导数、曲率和爱因斯坦方程变得奇异。然而，奇异度规时空几何学的最新进展表明，有方法可以将爱因斯坦方程和其他场方程解析地扩展到这种奇异性之外。这意味着信息可以从奇点中出来。在带电黑洞的情形下，所得解具有{ \nonsing }电磁场.另外，如果粒子是这样的黑洞，时空就会经历降维效应，就像某些微扰量子引力方法所要求的那样。

Quantum Geometric Confinement and Dynamical Transmission in Grushin Cylinder

We classify the self-adjoint realisations of the Laplace-Beltrami operator minimally defined on an infinite cylinder equipped with an incomplete Riemannian metric of Grushin type, in the class of metrics yielding an infinite deficiency index. Such realisations are naturally interpreted as Hamiltonians governing the geometric confinement of a Schrödinger quantum particle away from the singularity, or the dynamical transmission across the singularity. In particular, we characterise all physically meaningful extensions qualified by explicit local boundary conditions at the singularity. Within our…