Hermitian Yang-Mills Metrics on Higgs Bundles over Asymptotically Cylindrical Kähler Manifolds

Let V be an asymptotically cylindrical Kahler manifold with asymptotic cross-section     

On Compact Hermitian Manifolds with Flat Gauduchon Connections

Given a Hermitian manifold(M~n, g), the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call▽~s=(1-s/2)▽~c+s/2▽~b the s-Gauduchon connection of M, where ▽~c and ▽~b are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat s-Gauduchon connection. This is related to a question asked by Yau. The cases with s = 0(a flat Chern connection) or s = 2(a flat Bismut connection) are classified respectively by Boothby in the1950 s or by the authors in a recent joint work with Q. Wang. In this article, we observe that if either s ≥ 4 + 2×3~(1/2) ≈ 7.46 or s ≤ 4-2×3~(1/2) ≈ 0.54 and s ≠ 0, then g is K?hler. We also show that, when n = 2,g is always K?hler unless s = 2. Therefore non-K?hler compact Gauduchon flat surfaces are exactly isosceles Hopf surfaces.     

Flat Riemannian Manifolds as Torus Bundles

Vasquez showed that for any finite group G there exists a number n(G) such that for every flat Riemannian manifold M with holonomy group G there exists a fiber bundle T → M → N, where T is a flat torus and N is a flat manifold of dimension less than or equal to n(G). We show that n(H) ≤ n(G) if H Δ leftG or G = N ? H and use this result to describe groups with the Vasquez number equal to 2 or 3.     

Cohomogeneity Two Actions on Flat Riemannian Manifolds

In this paper, we study flat Riemannian manifolds which have codimension two orbits, under the action of a closed and connected Lie group G of isometries. We assume that G has fixed points, then characterize M and orbits of M.     

η-Invariant and Flat Vector Bundles

We present an alternate definition of the mod Z component of the AtiyahPatodi-Singer η invariant associated to (not necessary unitary) fiat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah, Parodi and Singer.     

A Higgs bundle on a Hermitian symmetric space

Let \(M := \Gamma\backslash G/K\) be the quotient of an irreducible Hermitian symmetric space G/K by a torsionfree cocompact lattice \(\Gamma\subset G\) . There is a natural flat principal G-bundle over the compact Kähler manifold M which is constructed from the principal Γ-bundle over M defined by the quotient map \(G/K\longrightarrow M\) . We construct the principal G-Higgs bundle over M corresponding to this flat G-bundle. This principal G-Higgs bundle is rigid if \({\rm dim}_\mathbb{C} M\,\geq\,2\) .     

Complete Flat Affine and Lorentzian Manifolds

Following in the tradition of Hilbert's 18th problem of classifying crystallographic groups, we provide a survey of a series of results which have culminated in the study of flat Lorentz manifolds. In particular, Milnor asked whether all complete flat affine manifolds have virtually polycyclic fundamental groups. Margulis answered this question negatively by constructing complete flat Lorentz manifolds with free fundamental groups. In this paper, we follow the effort to classify and understand these interesting counterexamples to Milnor's question, and their generalizations.     

A Note on Hermitian-Einstein Metrics on Parabolic Stable Bundles

Let ${\overline M}Let be a compact complex manifold of complex dimension two with a smooth K?hler metric and D a smooth divisor on . If E is a rank 2 holomorphic vector bundle on with a stable parabolic structure along D, we prove that there exista a Hermitian-Einstein metric on compatible with the parabolic structure, whose curvature is square integrable. Received February 18, 2000, Accepted September 6, 2000     

Hermitian Metric Rigidity for Foliated Vector Bundles

We show that the canonical isometric imbedding of the symplectic group Sp(n) into R⁴n² gives the least-dimensional isometric imbedding into the Euclidean space, even in the local standpoint. We prove this result by calculating the quantity p_G determined by the curvature of Sp(n), which serves as an obstruction to the existence of local isometric imbeddings. We also exhibit the estimates on the value p_G for the remaining compact classical simple Lie groups, and improve the previous results on the codimension of local isometric imbeddings of these groups.     

Dirichlet problem for Hermitian Yang-Mills-Higgs equations over Hermitian manifolds

In this paper, we investigate the Dirichlet problem for one type of vortex equations, which generalize the well-known Hermitian Yang-Mills equations, over general Hermitian manifolds.     

关于紧致广义凯莱流形上的希格斯场的一些注记

在本文中,我们首先解决在紧致广义凯莱流形上的$I_{+-}$-希格斯丛上的阿尔法-厄米特-杨-米尔斯-希格斯方程的狄利克雷边值问题.然后我们证明了在闭的广义凯莱流形上的$I_{+-}$-希格斯丛,它的阿尔法半稳定性蕴含了渐进$\alpha$-厄米特-杨-米尔斯-希格斯结构的存在性.     

完备Riemann流形具有闭的割空间

本文证明了完备的Ｒｉｅｍａｎｎ流形即拥有闭的割空间（ｃｕｔｓｐａｃｅ），这一结论不但完满解答了段海豹在［１］中提出的问题１．６，大大地改进了他的主要结果（［１］，定理１．３），而且作为一个推论，我们还得到了经典Ｂｏｒｓｕｋ－Ｕｌａｍ定理的一个进一步推广．     

Hermite流形上Coupled Vortex方程的Dirichlet问题

本文研究Coupled Vortex方程的Dirichlet问题,通过热流方法来讨论该方程Dirichlet问题的解的存在唯一性．     

Complete Systems of Lines on a Hermitian Surface over a Finite Field

The aim is to find the maximum size of a set of mutually ske lines on a nonsingular Hermitian surface in PG(3, q) for various values of q. For q = 9 such extremal sets are intricate combinatorial structures intimately connected ith hemisystems, subreguli, and commuting null polarities. It turns out they are also closely related to the classical quartic surface of Kummer. Some bounds and examples are also given in the general case.     

一般Hopf曲面上向量丛的结构

得到了非主Hopf曲面上连续复向量丛全纯结构的存在性及可滤性问题的充要条件.     

η-Invariant and Flat Vector Bundles

Abstract We present an alternate definition of the mod Z component of the Atiyah-Patodi-Singer η invariant associated to (not necessary unitary) flat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah, Patodi and Singer. * Project supported by the Cheung-Kong Scholarship of the Ministry of Education of China and the 973 Project of the Ministry of Science and Technology of China. (Dedicated to the memory of Shiing-Shen Chern)     

Geometry of the Second-Order Tangent Bundles of Riemannian Manifolds

Let (M,g) be an n-dimensional Riemannian manifold and T2M be its secondorder tangent bundle equipped with a lift metric (g).In this paper,first,the authors construct some Riemannian almost product structures on (T2M,(g)) and present some results concerning these structures.Then,they investigate the curvature properties of (T2M,(g)).Finally,they study the properties of two metric connections with nonvanishing torsion on (T2 M,(g)):The H-lift of the Levi-Civita connection of g to T2 M,and the product conjugate connection defined by the Levi-Civita connection of (g) and an almost product structure.     

关于近Hermitian流形的注记(英文)

We give a necessary and sufficient condition for an almost Hermitian manifold to be a Kahler manifold. By making use of this condition, we give a new proof of Goldberg＇s theorem.     

Hessian Comparison and Spectrum Lower Bound of Almost Hermitian Manifolds

The authors obtain a complex Hessian comparison for almost Hermitian manifolds, which generalizes the Laplacian comparison for almost Hermitian manifolds by Tossati, and a sharp spectrum lower bound for compact quasi Kähler manifolds and a sharp complex Hessian comparison on nearly Kähler manifolds that generalize previous results of Aubin, Li Wang and Tam-Yu.