The gluing formula of the refined analytic torsion for an acyclic Hermitian connection

In the previous study by Huang and Lee (arXiv:1004.1753) we introduced the well-posed boundary conditions ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ and ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined analytic torsion for an acyclic Hermitian connection with respect to the boundary conditions ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ and ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ . In this case the refined analytic torsion consists of the Ray-Singer analytic torsion, the eta invariant and the values of the zeta functions at zero. We first compare the Ray-Singer analytic torsion and eta invariant subject to the boundary condition ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ or ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ with the Ray-Singer analytic torsion subject to the relative (or absolute) boundary condition and eta invariant subject to the APS boundary condition on a compact manifold with boundary. Using these results together with the well known gluing formula of the Ray-Singer analytic torsion subject to the relative and absolute boundary conditions and eta invariant subject to the APS boundary condition, we obtain the main result.     

Geometric BVPs, Hardy spaces, and the Cauchy integral and transform on regions with corners

In this paper we give a new perspective on the Cauchy integral and transform and Hardy spaces for Dirac-type operators on manifolds with corners of codimension two. Instead of considering Banach or Hilbert spaces, we use polyhomogeneous functions on a geometrically “blown-up” version of the manifold called the total boundary blow-up introduced by Mazzeo and Melrose [R.R. Mazzeo, R.B. Melrose, Analytic surgery and the eta invariant, Geom. Funct. Anal. 5 (1) (1995) 14-75]. These polyhomogeneous functions are smooth everywhere on the original manifold except at the corners where they have a “Taylor series” (with possible log terms) in polar coordinates. The main application of our analysis is a complete Fredholm theory for boundary value problems of Dirac operators on manifolds with corners of codimension two.     

Approximately invariant manifolds and global dynamics of spike states

We investigate the existence of a true invariant manifold given an approximately invariant manifold for an infinite-dimensional dynamical system. We prove that if the given manifold is approximately invariant and approximately normally hyperbolic, then the dynamical system has a true invariant manifold nearby. We apply this result to reveal the global dynamics of boundary spike states for the generalized Allen–Cahn equation.     

A property that characterizes Euler characteristic among invariants of combinatorial manifolds

If a real valued invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension in the manifold, then the invariant is completely determined by the Euler characteristic of the manifold and its boundary. So essentially, the Euler characteristic is the unique invariant of this type.     

Homology of pseudodifferential operators on manifolds with fibered cusps

The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals that generate the -dimensional Hochschild cohomology groups, the index of a fully elliptic fibered cusp operator is expressed as the sum of a local contribution of Atiyah-Singer type and a global term on the boundary. We announce a result relating this boundary term to the adiabatic limit of the eta invariant in a particular case.

     

The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group \mathbbZ_p{\mathbb{Z}_p}, where p is an odd prime. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p = n = 3. We use the expressions obtained to show that any such manifold is trivial in the appropriate reduced equivariant spin bordism group.     

Kojima's eta-function for manifold links in higher dimensions

Kojima's -function is generalized to give a new concordance invariant for certain two-component manifold links in higher dimensions. Examples are given of manifold links successfully distinguished by this generalized -function but not by their Cochran derived invariants.

     

Relative index pairing and odd index theorem for even dimensional manifolds

We prove an analogue for even dimensional manifolds of the Atiyah-Patodi-Singer twisted index theorem for trivialized flat bundles. We show that the eta invariant appearing in this result coincides with the eta invariant by Dai and Zhang up to an integer. We also obtain the odd dimensional counterpart for manifolds with boundary of the relative index pairing by Lesch, Moscovici and Pflaum.     

On the gluing problem for Dirac operators on manifolds with cylindrical ends

Combining elements of the b-calculus and the theory of elliptic boundary value problems, we solve the gluing problem for b-determinants of Dirac type operators on manifolds with cylindrical ends. As a corollary of our proof, we derive a gluing formula for the b-eta invariant and also a relative invariant formula relating the b-spectral invariants on a manifold with cylindrical end to the spectral invariants with the augmented APS boundary condition on the corresponding compact manifold with boundary.     

Eta invariant and holonomy: The even dimensional case

In previous work, we introduced eta invariants for even dimensional manifolds. It plays the same role as the eta invariant of Atiyah–Patodi–Singer, which is for odd dimensional manifolds. It is associated to K¹$K^1$ representatives on even dimensional manifolds and is closely related to the so called WZW theory in physics. In fact, it is an intrinsic interpretation of the Wess–Zumino term without passing to the bounding 3-manifold. Spectrally the eta invariant is defined on a finite cylinder, rather than on the manifold itself. Thus it is an interesting question to find an intrinsic spectral interpretation of this new invariant. We address this issue here using adiabatic limit technique. The general formulation relates the (mod Z$\mathbb{Z}$ reduction of) eta invariant for even dimensional manifolds with the holonomy of the determinant line bundle of a natural family of Dirac type operators. In this sense our result might be thought of as an even dimensional analogue of Witten's holonomy theorem proved by Bismut–Freed and Cheeger independently.     

Regularity theory for the generalized Neumann problem for Yang-Mills Connections – Non-trivial examples in dimensions 3 and 4

We develop the existence and regularity theory for the generalized Neumann problem for Yang-Mills connections. This is the most general boundary value problem for connections on a compact manifold with smooth boundary, with geometric meaning. It is obtained by reflecting the base manifold across its boundary and lifting this action non-trivially to the bundle. The prescribed lifting corresponds to a geometric invariant, which is similar to the monopole number. When this invariant is non-zero, there exist non-trivial solutions of the generalized Neumann problem. We prove the existence of non-trivial solutions over the 3-dimensional disk and over the 4-dimensional manifold We outline the procedure for finding non-trivial examples of solutions over more general manifolds of dimension 3 and 4. Received: 20 June 1999     

Positive scalar curvature,higher rho invariants and localization algebras

In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac operator on a spin manifold with boundary to the higher rho invariant of the Dirac operator on the boundary, where the boundary is endowed with a positive scalar curvature metric. Our result extends a theorem of Piazza and Schick [27, Theorem 1.17].     

Invariants of Isospectral Deformations and Spectral Rigidity

We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace–Beltrami operator on a compact Riemannian manifold with Robin boundary conditions. Given a Kronecker invariant torus Λ of the billiard ball map with a Diophantine vector of rotation we prove that certain integrals on Λ involving the function in the Robin boundary conditions remain constant under weak isospectral deformations. To this end we construct continuous families of quasimodes associated with Λ. We obtain also isospectral invariants of the Laplacian with a real-valued potential on a compact manifold for continuous deformations of the potential. These invariants are obtained from the first Birkhoff invariant of the microlocal monodromy operator associated to Λ. As an application we prove spectral rigidity of the Robin boundary conditions in the case of Liouville billiard tables of dimension two in the presence of a (?/2?)² group of symmetries.     

The index of elliptic operators on manifolds with conical points

For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the "eta" invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be nonzero. Moreover, we introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.     

The Eta-Invariant and Pontryagin Duality in K-Theory

The topological significance of the spectral Atiyah--Patodi--Singer -invariant is investigated. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. Pontryagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented.     

弱双曲覆盖映射的单一化结构稳定性

本文引入了弱双曲不变集的定义，给出弱双曲不变集的一些性质，并证明了弱双曲覆盖映射是单一化结构稳定的．     

Partial synchronization in coupled chemical chaotic oscillators

In this paper we investigate the problem of partial synchronization in diffusively coupled chemical chaotic oscillators with zero-flux boundary conditions. The dynamical properties of the chemical system which oscillates with Uniform Phase evolution, yet has Chaotic Amplitudes (UPCA) are first discussed. By combining numerical and analytical methods, the impossibility of full global synchronization in a network of two or three coupled chemical oscillators is discovered. Mathematically, stable partial synchronization corresponds to convergence to a linear invariant manifold of the global state space. The sufficient conditions for exponential stability of the invariant manifold in a network of three coupled chemical oscillators are obtained via the nonlinear contraction principle.     

An Intrinsic Geometric Formulation of the Equilibrium Equations in Continuum Mechanics

The theory of invariant continuum mechanics is based on the concept that forces and stresses are defined as elements of the cotangent bundle of the configuration manifold. While body and physical space are modeled as differentiable manifolds, the infinite dimensional configuration manifold is given by all configurations of the body in the physical space. In this paper a virtual work principle is postulated which leads together with an induced traction stress and Stokes' theorem directly to the local equilibrium equations and the traction boundary conditions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rough isometry and energy finite solutions of elliptic equations on Riemannian manifolds

We prove that if the s-harmonic boundary of a complete Riemannian manifold consists of finitely many points, then the set of bounded energy finite solutions for certain nonlinear elliptic operators on the manifold is one to one corresponding to , where l is the cardinality of thes-harmonic boundary. We also prove that the finiteness of cardinality of s-harmonic boundary is a rough isometric invariant, moreover, in this case, the cardinality is preserved under rough isometries between complete Riemannian manifolds. This result generalizes those of Yau, of Donnelly, of Grigor'yan, of Li and Tam, of Kim and the present author, of Holopainen, and of the present author, but with different techniques which are demanded by the peculiarity of nonlinearity. Received October 13, 1999 / Revised November 23, 1999 / Published online July 20, 2000