Connections and Higgs fields on a principal bundle

Let M be a compact connected Kähler manifold and G a connected linear algebraic group defined over \({\mathbb{C}}\) . A Higgs field on a holomorphic principal G-bundle ε _G over M is a holomorphic section θ of \(\text{ad}(\epsilon_{G})\otimes {\Omega}^{1}_{M}\) such that θ∧ θ = 0. Let L(G) be the Levi quotient of G and (ε _G (L(G)), θ _l ) the Higgs L(G)-bundle associated with (ε _G , θ). The Higgs bundle (ε _G , θ) will be called semistable (respectively, stable) if (ε _G (L(G)), θ _l ) is semistable (respectively, stable). A semistable Higgs G-bundle (ε _G , θ) will be called pseudostable if the adjoint vector bundle ad(ε _G (L(G))) admits a filtration by subbundles, compatible with θ, such that the associated graded object is a polystable Higgs vector bundle. We construct an equivalence of categories between the category of flat G-bundles over M and the category of pseudostable Higgs G-bundles over M with vanishing characteristic classes of degree one and degree two. This equivalence is actually constructed in the more general equivariant set-up where a finite group acts on the Kähler manifold. As an application, we give various equivalent conditions for a holomorphic G-bundle over a complex torus to admit a flat holomorphic connection.     

一个常微分方程整体解的存在性结果

研究常微分方程d2u/dx2 K(x)e2u=0在(-∞, ∞)上整体解的存在性问题.此方程是熟知的在R2上预定高斯曲率方程的一个特例.本文证明了一个存在性定理.     

Exotic Expansions and Pathological Properties of <Emphasis Type="Italic">ζ</Emphasis>-Functions on Conic Manifolds

We give a complete classification and present new exotic phenomena of the meromorphic structure of ζ-functions associated to general self-adjoint extensions of Laplace-type operators over conic manifolds. We show that the meromorphic extensions of these ζ-functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. The corresponding heat kernel and resolvent trace expansions also exhibit exotic behaviors with logarithmic terms of arbitrary positive and negative multiplicity. We also give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.     

Cobordism,Relative Indices and Stein Fillings

In this article we build on the framework developed in Ann. Math. 166, 183–214 ([2007]), 166, 723–777 ([2007]), 167, 1–67 ([2008]) to obtain a more complete understanding of the gluing properties for indices of boundary value problems for the Spin _ℂ-Dirac operator with sub-elliptic boundary conditions. We extend our analytic results for sub-elliptic boundary value problems for the Spin _ℂ-Dirac operator, and gluing results for the indices of these boundary problems to Spin _ℂ-manifolds with several pseudoconvex (pseudoconcave) boundary components. These results are applied to study Stein fillability for compact, 3-dimensional, contact manifolds. This material is based upon work supported by the National Science Foundation under Grant No. 0603973, and the Francis J. Carey term chair. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.     

VANISHING THEOREMS FOR ACH K(A)HLER MANIFOLDS AND HARMONIC MAPS

We discuss a class of complete Kaler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L2 cohomology of such manifolds when it has positive spectrum. We also generalize the result the weighted function p is of sub-quadratic growth of the distance function. We also obtain inequality.     

Analytic representations in the three-dimensional Frobenius problem

We consider the Diophantine problem of Frobenius for the semigroup , where d ³ denotes the triple (d ₁,d ₂,d ₃), gcd (d ₁,d ₂,d ₃)=1. Based on the Hadamard product of analytic functions, we find the analytic representation of the diagonal elements a _kk (d ³) of Johnson’s matrix of minimal relations in terms of d ₁, d ₂, and d ₃. With our recent results, this gives the analytic representation of the Frobenius number F(d ³), genus G(d ³), and Hilbert series H(d ³;z) for the semigroups . This representation complements Curtis’s theorem on the nonalgebraic representation of the Frobenius number F(d ³). We also give a procedure for calculating the diagonal and off-diagonal elements of Johnson’s matrix.      

On Surface Attractors and Repellers in 3-Manifolds

We show that if f: M ³ → M ³ is an A diffeomorphism with a surface two-dimensional attractor or repeller $\mathcal{B}$ with support $M_\mathcal{B}^2$ , then $\mathcal{B} = M_\mathcal{B}^2$ and there exists a k ≥ 1 such that (1) $M_\mathcal{B}^2$ is the disjoint union M ₁ ² ? ? ? M _k ² of tame surfaces such that each surface M _i ² is homeomorphic to the 2-torus T ²; (2) the restriction of f ^k to M _i ² , i ∈ {1,..., k}, is conjugate to an Anosov diffeomorphism of the torus T ².     

On discretizations of invariant foliations over inertial manifolds

Invariant foliations over inertial manifolds of partial differential equations under numerical discretizations are studied. It is proved that the numerical method considered as a discrete dynamical system has C¹-close invariant foliations. The rate of the C¹-convergence is estimated as well.     

PLEMELJ FORMULA OF CAUCHY TYPE INTEGRAL ON STEIN MANIFOLDS

Let D be a bounded domain with piecewise C(1) smooth orientable boundary on Stein manifolds, and let Φ(z) be a Cauchy type integral with Bochner-Martinelli kernel Ω(ψv, -S, S) and Holder continuous density function ψ(ζ), the authors define a solid angular coefficient α(t) at the point t ∈(e)D, prove that there exist the interior and outer limit values Φ± (t) under the meaning of the Cauchy principal value, and obtain the more general Plemelj formula and jump formula.