流形 X 的 Riesz z-能量是两点间距离的 z 次幂在乘积空间 X×X 上的积分。把它视为复变量 z 的函数时，可通过解析延拓推广为亚纯函数，我们称之为 X 的亚纯能量函数。它仅在某些负整数处具有单极点。流形 X 的留数就是该亚纯能量函数的留数。例如，R³ 中曲面的体积和 Willmore 能量都可作为留数得到。本文中，我们首先证明欧几里得空间中闭子流形或紧体在 z=-2 dim X 处留数的 Möbius 不变性。我们引入紧体的相对留数和加权留数，并证明维数小于 4 的紧体的标量曲率、平均曲率以及 Euler 示性数都可用留数和局部留数表示。当 X 是欧几里得空间中的闭子流形时，我们研究为获得（全局）留数而对 X 的局部定义函数所需的微分阶数。我们还证明了容斥原理。留数看起来类似于通过渐近展开得到的量，例如内蕴体积（Lipschitz-Killing 曲率）、Laplacian 谱以及 Graham-Witten 能量。我们证明留数独立于这些量。最后，我们为 R⁵ 中的 4 维超曲面引入一种 Möbius 不变主曲率能量，并将 Graham-Witten 能量用留数、Weyl 张量以及这种 Möbius 不变主曲率能量表示。
英文摘要：
The Riesz $z$-energy of a manifold $X$ is the integration of the distance between two points to the power $z$ over the product space $X\times X$. Considered as a function of a complex variable $z$, it can be generalized to a meromorphic function by analytic continuation, which we will call the meromorphic energy function of $X$. It has only simple poles at some negative integers. The residues of a manifold $X$ are the residues of the meromorphic energy function. For example, the volume and the Willmore energy for surfaces in $\mathbb{R}^3$ can be obtained as residues.
In this paper we first show the Möbius invariance of the residue at $z=-2\dim X$ of a closed submanifold or a compact body in a Euclidean space. We introduce the relative residues for compact bodies and weighted residues, and show that the scalar curvature and the mean curvature as well as the Euler characteristic of compact bodies of dimension less than $4$ can be expressed in terms of residues and local residues. We study the order of differentiaion of a local defining function of $X$ that is necessary to obtain the (global) residues when $X$ is a closed submanifold of a Euclidean space. We also show the inclusion-exclusion principle.
Residues appear to be similar to quantities obtained by asymptotic expansion such as intrinsic volumes (Lipschitz-Killing curvatures), spectra of Laplacian, and the Graham-Witten energy. We show that residues are independent from them. Finally we introduce a \M invariant principal curvature energy for $4$-dimensional hypersurfaces in $\mathbb{R}^5$, and express the Graham-Witten energy in terms of the residues, Weyl tensor, and this Möbius invariant principal curvature energy.
This note is to concern a generalization to the case of twisted coefficients of the classical theory of Abelian differentials on a compact Riemann surface. We apply the Dirichlet's principle to a modified energy functional to show the existence of differentials with twisted coefficients of the second and third kinds under a suitable assumption on residues.
This paper review one construction of Frobenius manifolds (and slightly weaker structures). It splits it into several steps and discusses the freedom and the constraints in these steps. The steps pass through holomorphic bundles with meromorphic connections. A conjecture on existence and uniqueness of certain such bundles, a proof of the conjecture in the 2-dimensional cases, and some other new results form a research part of this paper.
We study the local Killing Lie algebra of meromorphic almost rigid geometric structures on complex manifolds. This leads to classification results for compact complex manifolds bearing holomorphic rigid geometric structures.
The base space of a semi-universal unfolding of a hypersurface singularity carries a rich geometric structure, which was axiomatized as a CDV-structure by C. Hertling. For any CDV-structure on a Frobenius manifold M, the pull-back of the (1,0)-tangent bundle of M to the product of M by the complex line carries two natural holomorphic structures equipped with flat meromorphic connections. We show that, for any semi-simple CDV-structure, there is a formal isomorphism between these two bundles compatible with connections. Moreover, if we assume that the super-symmetric index Q vanishes, we give a…
This are the notes of a course, given by the first author for the Graduiertenkollegs (=graduate students) at the Ruhr-University Bochum, in December 1997. These lectures pursued two main tasks: FIRST - to give a systematic and self-contained introduction to the Gromov theory of pseudoholomorphic curves. This is done in Chapters I,II,III. SECOND - to explain our join results on envelopes of meromorphy of real surfaces in complex two-dimensional manifolds. We do this in Chapter IV.
In this paper, we define the spectral Einstein functional associated with the Dirac operator for manifolds with boundary. And we give the proof of Kastler-Kalau-Walze type theorem for the spectral Einstein functional associated with the Dirac operator on 4-dimensional manifolds with boundary.
We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian $Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their Poincaré-Einstein metrics $(X,g_+)$. Our approach is based on the spectral theory of Dirac operator in the ambient $Spin$-manifold, and associated spinor valued meromorphic family of distributions with residues given by the residue family operators $\slashed{D}_N^{res}(h;λ)$ on spinors. We develop basic aspects and properties of $\slashed{D}_N^{res}(h;λ)$ including conformal covariance, factorization properties by conformally covariant operators…
In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.
In this paper, we prove a Kastler-Kalau-Walze type theorem for 4-dimensional and 6-dimensional spin manifolds with boundary associated with the conformal Robertson-Walker metric. And we give two kinds of operator theoretic explanations of the gravitational action for boundary in the case of 4-dimensional manifolds with flat boundary. In particular, for 6-dimensional spin manifolds with boundary with the conformal Robertson-Walker metric, we obtain the noncommutative residue of the composition of $π^+D^{-1}$ and $π^+D^{-3}$ is proportional to the Einstein-Hilbert action for manifolds with bound…
In this paper we give formulae for the Dixmier trace and the noncommutative residue (also called Wodzicki's residue) of pseudo-differential operators by using the notion of global symbol. We consider both cases, compact manifolds with or without boundary. Our analysis on the Dixmier trace of invariant pseudo-differential operators on closed manifolds will be based on the Fourier analysis associated to every elliptic and positive operator and the quantization process developed by Delgado and Ruzhansky. In particular, for compact Lie groups this can be done by using the representation theory of …
The Ray-Singer analytic torsion is the zeta-function trace of a certain sum of logarithm operators on the de Rham complex. In this note we examine the residue analytic torsion, defined using the residue-trace instead of the spectral zeta function quasi-trace.
In this paper, for an even dimensional compact manifold with boundary which has the non-product metric near the boundary, we use the noncommutative residue to define a conformal invariant pair. For a 4-dimensional manifold, we compute this conformal invariant pair under some conditions and point out the way of computations in the general.
An analogue of Brylinski's knot beta function is defined for a compactly supported (Schwartz) distribution $T$ on $d$-dimensional Euclidean space. This is a holomorphic function on a right half-plane. If $T$ is a (uniform) double-layer on a compact smooth hypersurface, then the beta function has an analytic continuation to the complex plane as a meromorphic function, and the residues are integrals of invariants of the second fundamental form. The first few residues are computed when $d=2$ and $d=3$.
We prove a Kastler-Kalau-Walze type theorem for the Dirac operator and the signature operator for $3,4$-dimensional manifolds with boundary. As a corollary, we give two kinds of operator theoretic explanations of the gravitational action in the case of 4-dimensional manifolds with flat boundary.
In [3], Connes found a conformal invariant using Wodzicki's 1-density and computed it in the case of 4-dimensional manifold without boundary. In [14], Ugalde generalized the Connes' result to $n$-dimensional manifold without boundary. In this paper, we generalize the results of [3] and [14] to the case of manifolds with boundary.
An analogue of Brylinski's knot beta function is defined for a submanifold of d-dimensional Euclidean space. This is a meromorphic function on the complex plane. The first few residues are computed for a surface in three dimensional space.
In this paper, we give two Lichnerowicz type formulas for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection. We also prove two Kastler-Kalau-Walze type theorems for twisted Dirac operators and twisted signature operators on 4-dimensional manifolds with (resp. without) boundary.
We review previous work of Alain Connes, and its extension by the author, on some conformal invariants obtained from the noncommutative residue on even dimensional compact manifolds without boundary. Inspired by recent work of Yong Wang, we also address possible generalizations of these conformal invariants to the setting of compact manifolds with boundary.
For compact real manifolds, a new double conformal invariant is constructed using the Wodzicki residue and the $d$ operator in the framework of Connes. In the flat case, we compute this double conformal invariant, and in some special cases, we also compute this double conformal invariants. For complex manifolds, a new double conformal invariant is constructed using the Wodzicki residue and the $\bar{\partial}$ operator in the same way, and this double conformal invariant is computed in the flat case.
For an even dimensional, compact, conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold. In the conformally flat case, this operator coincides with the critical {\sf GJMS} operator of Graham-Jenne-Mason-Sparling. We use the Wodzicki residue of a pseudo-differential operator of order $-2,$ originally defined by A. Connes, acting on middle dimension forms.
We introduce the beta function of a knot in euclidean three-space. This is a meromorphic function of a complex variable which we prove admits a Bernstein type functional equation. We determine the first residues.
In this paper we prove geometric residue theorems for bundle maps over a compact manifold. The theory developed associates residues to the singularity submanifolds of the map for any invariant polynomial. The theory is then applied to a variety of settings: smooth maps between equidimensional manifolds, CR-singularities, finite singularities and singularities of odd forms as spinor bundle maps.
In this paper we produce several new invariants for CR and contact manifolds by looking at the noncommutative residue traces of various geometric projections. In the CR setting these operators arise from the Kohn-Rossi complex and include the Szegö projections on forms. In the contact setting they stem from the generalized Szegö projections at arbitrary integer levels of Epstein-Melrose and from the contact complex of Rumin. In particular, we recover and extend recent results of Hirachi and Boutet de Monvel and answer a question of Fefferman.
In this note, we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds with variable negative curvature. Applying results from dynamics on these spaces, we obtain optimal meromorphic extensions of weighted dynamical zeta functions and asymptotic counting estimates for the number of weighted closed geodesics. A meromorphic extension of the standard dynamical zeta function and the prime orbit theorem follow as corollaries. Finally…
An old theorem of Weil and Kodaira says that for a compact Kähler manifold $X$ there is a closed logarithmic $1$-form with residue divisor $D$ if and only if $D$ is homologous to zero in $H_{2n-2}(X,\mathbb C)$. In the first part of this paper, we generalize the above theorem to general compact complex manifolds by showing that the necessary and sufficient condition in general is described by a holomorphic invariant called the $\mathcal Q$-flat class. Next, we prove that the holomorphic criterion is reduced to the topological one when $X$ has Property $(H)$. Since all Kähler manifolds have Pro…
In this paper, we get a Kastler-Kalau-Walze type theorem associated to nonminimal de Rham-Hodge operators on compact manifolds with boundary. We give two kinds of operator-theoretic explanations of the gravitational action in the case of four dimensional compact manifolds with flat boundary.
We study the meromorphic open-string vertex algebras and their modules over the two-dimensional Riemannian manifolds that are complete, connected, orientable, and of constant sectional curvature $K\neq 0$. Using the parallel tensors, we explicitly determine a basis for the meromorphic open-string vertex algebra, its modules generated by eigenfunctions of the Laplace-Beltrami operator, and their irreducible quotients. We also study the modules generated by lowest weight subspace satisfying a geometrically interesting condition. It is showed that every irreducible module of this type is generate…
Let $M$ be a smooth manifold and $G$ a compact connected Lie group acting on $M$ by isometries. In this paper, we study the equivariant cohomology of ${\bf X}=T^\ast M$, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae. In case that $\bf X$ is a compact symplectic manifold with a Hamiltonian $G$-action, similar residue formulae were derived by Jeffrey, Kirwan et al.
This paper has four main parts. In the first part, we construct a noncommutative residue for the hypoelliptic calculus on Heisenberg manifolds, that is, for the class of Heisenberg PsiDOs introduced by Beals-Greiner and Taylor. This noncommutative residue appears as the residual trace on integer order Heisenberg PsiDOs induced by the analytic extension of the usual trace to non-integer order Heisenberg PsiDOs. Moreover, it agrees with the integral of the density defined by the logarithmic singularity of the Schwartz kernel of the corresponding Heisenberg PsiDO. In addition, we show that this n…
The goal of this article is to provide an explicit algorithmic construction of formal $F$-manifold structures, formal Frobenius manifold structures, and higher residue pairings on the primitive middle-dimensional cohomology $\mathbb{H}$ of a smooth projective Calabi-Yau complete intersection variety $X$ defined by homogeneous polynomials $G_1(\underline x), \dots, G_k(\underline x)$. Our main method is to analyze a certain dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra $\mathcal{A}$ obtained from the twisted de Rham complex which computes $\mathbb{H}$. More explicitly, we introduc…
We study the Hochschild homology groups of the algebra of complete symbols on a foliated manifold $(M,F)$. The first step is to relate these groups to the Poisson homology of $(M,F)$ and of other related foliated manifolds. We then establish several general properties of the Poisson homology groups of foliated manifolds. As an example, we completely determine these Hochschild homology groups for the algebra of complete symbols on the irrational slope foliation of a torus (under some diophantine approximation assumptions). We also use our calculations to determine all residue traces on algebras…
In this paper, we study the interplay between modules and sub-objects in holomorphic Poisson geometry. In particular, we define a new notion of "residue" for a Poisson module, analogous to the Poincaré residue of a meromorphic volume form. Of particular interest is the interaction between the residues of the canonical line bundle of a Poisson manifold and its degeneracy loci---where the rank of the Poisson structure drops. As an application, we provide new evidence in favour of Bondal's conjecture that the rank \leq 2k locus of a Fano Poisson manifold always has dimension \geq 2k+1. In particu…
For a pseudodifferential operator $S$ of order 0 acting on sections of a vector bundle $B$ on a compact manifold $M$ without boundary, we associate a differential form of order dimension of $M$ acting on $C^\infty(M)\times C^\infty(M)$. This differential form $Ω_{n,S}$ is given in terms of the Wodzicki 1-density $\wres([S,f][S,h])$. In the particular case of an even dimensional, compact, conformal manifold without boundary, we study this differential form for the case $(B,S)=(\cH,F)$, that is, the Fredholm module associated by A. Connes to the manifold $M.$ We give its explicit expression in t…
Given a compatible vector field on a compact connected almost-complex manifold, we show in this article that the multiplicities of eigenvalues among the zero point set of this vector field have intimate relations. We highlight a special case of our result and reinterpret it as a vanishing-type result in the framework of the celebrated Atiyah-Bott-Singer localization formula. This new point of view, via the Chern-Weil theory and a strengthened version of Bott's residue formula observed by Futaki and Morita, can lead to an obstruction to Killing real holomorphic vector fields on compact Hermitia…
We extend Vasy's results on semiclassical high energy estimates for the meromorphic continuation of the resolvent for asymptotically hyperbolic manifolds to metrics that are not necessarily even. Vasy's method gives the meromorphic continuation of the resolvent and high energy estimates in strips, assuming that the geodesic flow is non-trapping, without having to construct a parametrix. We prove that the size of the strip to which the resolvent can be extended meromorphically is the same obtained by Guillarmou.
