SUB-SIGNATURE OPERATORS, η-INVARIANTS AND A RIEMANN-ROCH THEOREM FOR FLAT VECTOR BUNDLES

The author presents an extension of the Atiyah-Patodi-Singer invariant for unitary representations [2, 3] to the non-unitary case, as well as to the case where the base manifold admits certain finer structures. In particular, when the base manifold has a fibration structure, a Riemann-Roch theorem for these invariants is established by computing the adiabatic limits of the associated η-invariants.     

SUB-SIGNATURE OPERATORS,η-INVARIANTS AND A RIEMANN-ROCH THEOREM FOR FLAT VECTOR BUNDLES

§0. Introduction Let M be an odd dimensional closed oriented manifold. Let ρbe a unitary represen-tation of the fundamental group of M. By using their η-invariant, Atiyah-Patodi-Singer [2,3] introduced an R/Z-valued smooth invariant associated to …     

Laplacians and Spectrum for Singular Foliations

The author surveys Connes＇ results on the longitudinal Laplace operator along a （regular） foliation and its spectrum, and discusses their generalization to any singular foliation on a compact manifold. Namely, it is proved that the Laplacian of a singular foliation is an essentially self-adjoint operator （unbounded） and has the same spectrum in every （faithful） representation, in particular, in L2 of the manifold and L2 of a leaf. The author also discusses briefly the relation of the Baum-Connes assembly map with the calculation of the spectrum.     

Evolution and monotonicity of eigenvalues under the Ricci flow

Let(M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. We derive the evolution equation for the eigenvalues of geometric operator-△φ+ c R under the Ricci flow and the normalized Ricci flow, where △φis the Witten-Laplacian operator, φ∈ C∞(M), and R is the scalar curvature with respect to the metric g(t). As an application, we prove that the eigenvalues of the geometric operator are nondecreasing along the Ricci flow coupled to a heat equation for manifold M with some Ricci curvature condition when c 14.     

ORTHOGONAL POLYNOMIALS ASSOCIATED WITH THE DIRAC OPERATOR IN EUCLIDEAN SPACE

The author considers the possibility of generalizing the theory of classicalpolynomials to the higher dimensional case.The starting point is the splitting up ofthe second order differential operator of these polynomials into the derivation operator,considered as an operator between Hilbert spaces and its adjoint.In the case of severaldimensions the derivation operator is replaced by the Dirac operator.As however theset of polynomials in the vector variable x is not dense in the Hilbert modulesconsidered,first a decomposition of these modules in terms of spherical monogenicfunctions is proved.Then by applying the theory to each of the constituents,generalizations of the Gegenbauer and the Hermite polynomials are obtained.     

A Criterion of Nonparabolicity by the Ricci Curvature

A complete manifold is said to be nonparabolic if it does admit a positive Green’s function. To ?nd a sharp geometric criterion for the parabolicity/nonparbolicity is an attractive question inside the function theory on Riemannian manifolds. This paper devotes to proving a criterion for nonparabolicity of a complete manifold weakened by the Ricci curvature. For this purpose, we shall apply the new Laplacian comparison theorem established by the ?rst author to show the existence of a non-constant bounded subharmonic function.     

Eigenvalues of Spinc Dirac Operator

In this paper we research the lower bound of the eigenvalue of Spinc Dirac operator on the Spinc manifold. By the Weisenbock formula, we get an estimate of it, then following the idea of Th Friedrich [2] and X Zhang [6]. We get a finer estimate of it. As an application, we give a condition when the Seiberg-Witten equation only has 0 solution.     

Complex geometry and real geometry

There is a well-known way to generalize the Riemann-Roch operator for Kahler manifold to that for Hermitian manifold. In this paper we show a slightly different way to get a generalized Riemann-Roch operator, which is just the Dirac operator. The difference between the two operators is that the latter one enables the so-called Pythagoras equalities.     

Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds

Let M be a complete Riemannian manifold possibly with a boundary?M.For any C~1-vector field Z,by using gradient/functional inequalities of the(reflecting)diffusion process generated by L:=?+Z,pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of?M if it exists.These characterizations extend and strengthen the recent results derived by Naber for the uniform norm‖RicZ‖∞on manifolds without boundaries.A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first author,such that the proofs are significantly simplified.     

Inequalities of Eigenvalues for the Dirac Operator on Compact Complex Spin Submanifolds in Complex Projective Spaces

For a compact complex spin manifold M with a holomorphic isometric embed- ding into the complex projective space,the authors obtain the extrinsic estimates from above and below for eigenvalues of the Dirac operator,which depend on the data of an isometric embedding of M.Further,from the inequalities of eigenvalues,the gaps of the eigenvalues and the ratio of the eigenvalues are obtained.     

η-INVARIANT AND CHERN-SIMONS CURRENT

The author presents an alternate proof of the Bismut-Zhang localization formula of ηinvariants, when the target manifold is a sphere, by using ideas of mod k index theory instead of the difficult analytic localization techniques of Bismut-Lebeau. As a consequence, it is shown that the R/Z part of the aualytically defined η invariant of Atiyah-Patodi-Singer for a Dirac operator on an odd dimensional closed spin manifold can be expressed purely geometrically through a stable Chern-Simons current on a higher dimensional sphere. As a preliminary application, the author discusses the relation with the Atiyah-Patodi-Singer R/Z index theorem for unitary flat vector bundles,and proves an R refinement in the case where the Dirac operator is replaced by the Signature operator.     

Continuity of Dirac spectra

It is well known that on a bounded spectral interval the Dirac spectrum can be described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article, we extend this result to a global version. We view the spectrum of a Dirac operator as a function $\mathbb Z \,\rightarrow \mathbb R \,$ and endow the space of all spectra with an $\mathrm{arsinh }$ -uniform metric. We prove that the spectrum of the Dirac operator depends continuously on the Riemannian metric. As a corollary, we obtain the existence of a non-decreasing family of functions on the space of all Riemannian metrics, which represents the entire Dirac spectrum at any metric. We also show that, due to spectral flow, these functions do not descend to the space of Riemannian metrics modulo spin diffeomorphisms in general.     

The spectrum of twisted Dirac operators on compact flat manifolds

Let be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of , and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group , we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the -series, in terms of values of Hurwitz zeta functions, and the -invariant. We give the dimension of the space of harmonic spinors and characterize all -manifolds having asymmetric Dirac spectrum.

Furthermore, we exhibit many examples of Dirac isospectral pairs of -manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat -manifolds, pairwise nonhomeomorphic to each other of the order of .

     

Transverse homoclinic orbit bifurcated from a homoclinic manifold by the higher order melnikov integrals

Consider an autonomous ordinary differential equation in $\mathbb{R}^n$ that has a $d$ dimensional homoclinic solution manifold $W^H$. Suppose the homoclinic manifold can be locally parametrized by $(\alpha,\theta) \in \mathbb{R}^{d-1}\times \mathbb{R}$. We study the bifurcation of the homoclinic solution manifold $W^H$ under periodic perturbations. Using exponential dichotomies and Lyapunov-Schmidt reduction, we obtain the higher order Melnikov function. For a fixed $(\alpha_0,\theta_0)$ on $W^H$, if the Melnikov function have a simple zeros, then the perturbed system can have transverse homoclinic solutions near $W^H$.     

$R^n$空间中的Cauchy积分公式

In this note p(D) = Dm+ b1Dm 1+···+ bmis a polynomial Dirac operator in R~n, where D =nj=1ej xjis a standard Dirac operator in Rn, bjare the complex constant coefficients. In this note we discuss all decompositions of p(D) according to its coefficients bj,and obtain the corresponding explicit Cauchy integral formulae of f which are the solution of p(D)f = 0.     

单位方体上沿曲面的振荡积分在Sobolev空间上的有界性

研究了欧氏空间R~2中单位方体Q~2=[0,1]~2上沿曲面(t,s,γ(t,s))的振荡奇异积分算子T_(α,β)f(u,v,x)=∫_(Q~2)f(u-t,v-s,x-γ(t,s))e~(it~(-β_1)s~(-β_2))t~(-1-α_1)s~(-1-α_2)dtds从Sobolev空间L_τ~p(R~(2+n))到L~p(R~(2+n))中的有界性,其中x∈R~n,(u,v)∈R~2,(t,s,γ(t,s))=(t,s,t~(P_1)s~(q_1),t~(p_2)s~(q_2),…,t~(p_n)s~(q_n))为R~(2+n)上一个曲面,且β_1α_1≥0,β_2α_20.这些结果推广和改进了R~3上的某些已知的结果.作为应用,得到了乘积空间上粗糖核奇异积分算子的Sobolev有界性.     

ESTIMATE ON LOWER BOUND OF THE FIRST EIGENVALUE OF A COMPACT RIEMANNIAN MANIFOLD

The author gives an optimum estimate of the first eigenvalue of a compact Riemannian manifold. It is shown that let M be a compact Riemannian manifold, then the first eigenvalue λ_1 of the Laplace operator of M satisfies α_1+max{0,-(n-1)K}≥π~2/d~2 where d is the diameter of M and (n-1)K is the negative lower bound of the Ricci curvature of M.     

The $${\bar\mu}$$-invariant of Seifert fibered homology spheres and the Dirac operator

We derive a formula for the [`(m)]{\bar\mu}-invariant of a Seifert fibered homology sphere in terms of the η-invariant of its Dirac operator. As a consequence, we obtain a vanishing result for the index of certain Dirac operators on plumbed 4-manifolds bounding such spheres.     

Existence of Dirac eigenvalues of higher multiplicity

In this article, we prove that on any compact spin manifold of dimension \(m \equiv 0,6,7 \mod 8\), there exists a metric, for which the associated Dirac operator has at least one eigenvalue of multiplicity at least two. We prove this by “catching” the desired metric in a subspace of Riemannian metrics with a loop that is not homotopically trivial. We show how this can be done on the sphere with a loop of metrics induced by a family of rotations. Finally, we transport this loop to an arbitrary manifold (of suitable dimension) by extending some known results about surgery theory on spin manifolds.