共查询到20条相似文献,搜索用时 33 毫秒
1.
In this paper, we solve the gluing problem for the ζ-determinant of a Dirac Laplacian. To do so, we develop a new approach to solve such problems which relies heavily on the theory of elliptic boundary problems, the analysis of the resolvent of the Dirac operator, and the introduction of an auxiliary model problem. Moreover, as a byproduct of our approach we obtain a new gluing formula for the eta invariant au gratis. 相似文献
2.
Paul Loya 《Journal of Differential Equations》2007,239(1):132-195
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. 相似文献
3.
Let M be a complete Riemannian manifold and D⊂M a smoothly bounded domain with compact closure. We use Brownian motion to study the relationship between the Dirichlet spectrum of D and the heat content asymptotics of D. Central to our investigation is a sequence of invariants associated to D defined using exit time moments. We prove that our invariants determine that part of the spectrum corresponding to eigenspaces which are not orthogonal to constant functions, that our invariants determine the heat content asymptotics associated to the manifold, and that when the manifold is a generic domain in Euclidean space, the invariants determine the Dirichlet spectrum. 相似文献
4.
Yoonweon Lee 《Journal of Geometric Analysis》2006,16(4):633-660
In this article we discuss the asymptotic expansions of the zeta-determinants of Dirac Laplacians on a compact manifold with
boundary when the boundary part is stretched. In [12] the author studied the same question under the assumption of no existence
of L2 - and extended L2 -solutions of Dirac operators when the boundary part is stretched to infinite length. Therefore, the results in this article
generalize those in [12]. Using the main results we obtain the formula describing the ratio of two zeta-determinants of Dirac
Laplacians with the APS boundary conditions associated with two unitary involutions σ1 and σ2 on ker B satisfying Gσi = -σi G. We also prove the adiabatic decomposition formulas for the zeta-determinants of Dirac Laplacians on a closed manifold
when the Dirichlet or the APS boundary condition is imposed on partitioned manifolds, which generalize the results in [10]
and [11]. 相似文献
5.
Jianxun Hu 《Compositio Mathematica》2001,125(3):345-352
In this paper, using the gluing formula of Gromov–Witten invariants for symplectic cutting developed by Li and Ruan, we established some relations between Gromov–Witten invariants of a semipositive symplectic manifold M and its blow-ups along a smooth surface. 相似文献
6.
This paper begins a series devoted to developing a general and practical theory of moving frames for infinite-dimensional
Lie pseudo-groups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan
forms and the Cartan structure equations for a pseudo-group. Our approach is completely explicit and avoids reliance on the
theory of exterior differential systems and prolongation.
The second paper [60] will apply these constructions in order to develop the moving frame algorithm for the action of the
pseudo-group on submanifolds. The third paper [61] will apply Gr?bner basis methods to prove a fundamental theorem on the
freeness of pseudo-group actions on jet bundles, and a constructive version of the finiteness theorem of Tresse and Kumpera
for generating systems of differential invariants and also their syzygies.
Applications of the moving frame method include practical algorithms for constructing complete systems of differential invariants
and invariant differential forms, classifying their syzygies and recurrence relations, analyzing invariant variational principles,
and solving equivalence and symmetry problems arising in geometry and physics. 相似文献
7.
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]. 相似文献
8.
Xiaohuan Mo 《Differential Geometry and its Applications》2009,27(1):7-14
One of fundamental problems in Finsler geometry is to establish some delicate equations between Riemannian invariants and non-Riemannian invariants. Inspired by results due to Akbar-Zadeh etc., this note establishes a new fundamental equation between non-Riemannian quantity H and Riemannian quantities on a Finsler manifold. As its application, we show that all R-quadratic Finsler metrics have vanishing non-Riemannian invariant H generalizing result previously only known in the case of Randers metric. 相似文献
9.
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. 相似文献
10.
Sergiu Moroianu 《Advances in Mathematics》2005,194(2):504-519
The cobordism invariance of the index on closed manifolds is reproved using the calculus Ψc of cusp pseudodifferential operators on a manifold with boundary. More generally, on a compact manifold with corners, the existence of a symmetric cusp differential operator of order 1 and of Dirac type near the boundary implies that the sum of the indices of the induced operators on the hyperfaces is null. 相似文献
11.
Xiliang WANG 《Frontiers of Mathematics in China》2021,16(4):1075
Using the degeneration formula, we study the change of Gromov-Witten invariants under blow-up for symplectic 4-manifolds and obtain a genus-decreasing relation of Gromov-Witten invariant of symplectic four manifold under blow-up. 相似文献
12.
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. 相似文献
13.
Nora Ganter 《Advances in Mathematics》2006,205(1):84-133
We generalize the definition of orbifold elliptic genus and introduce orbifold genera of chromatic level h, using h-tuples rather than pairs of commuting elements. We show that our genera are in fact orbifold invariants, and we prove integrality results for them. If the genus arises from an H∞-map into the Morava-Lubin-Tate theory Eh, then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold M in terms of the genus of M itself. Our formula is the p-typical analogue of the Dijkgraaf-Moore-Verlinde-Verlinde formula for the orbifold elliptic genus [R. Dijkgraaf et al., Elliptic genera of symmetric products and second quantized strings Comm. Math. Phys. 185(1) (1997) 197-209]. It depends only on h and not on the genus. 相似文献
14.
Xianzhe Dai 《Transactions of the American Mathematical Society》2002,354(1):107-122
We prove an adiabatic limit formula for the eta invariant of a manifold with boundary. The eta invariant is defined using the Atiyah-Patodi-Singer boundary condition and the underlying manifold is fibered over a manifold with boundary. Our result extends the work of Bismut-Cheeger to manifolds with boundary.
15.
We consider partial regularity for energy minimizing maps satisfying a partially free boundary condition. This condition takes
the form of the requirement that a relatively open subset of the boundary of the domain manifold be mapped into a closed submanifold
with non-empty boundary, contained in the target manifold. We obtain an optimal estimate on the Hausdorff dimension of the
singular set of such a map. Our result can be interpreted as regularity result for a vector-valued Signorini, or thin-obstacle,
problem. 相似文献
16.
Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology
Qusay S.A. Al-Zamil James Montaldi 《Topology and its Applications》2012,159(3):823-832
In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Λ on a compact Riemannian manifold M with boundary ∂M determines de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and the corresponding vector field XM on M, Witten defines an inhomogeneous coboundary operator dXM=d+ιXM on invariant forms on M. The main purpose is to adapt Belishev-Sharafutdinov?s boundary data to invariant forms in terms of the operator dXM in order to investigate to what extent the equivariant topology of a manifold is determined by the corresponding variant of the DN map. We define an operator ΛXM on invariant forms on the boundary which we call the XM-DN map and using this we recover the XM-cohomology groups from the generalized boundary data (∂M,ΛXM). This shows that for a Zariski-open subset of the Lie algebra, ΛXM determines the free part of the relative and absolute equivariant cohomology groups of M. In addition, we partially determine the ring structure of XM-cohomology groups from ΛXM. These results explain to what extent the equivariant topology of the manifold in question is determined by ΛXM. 相似文献
17.
Log Gromov-Witten invariants have recently been defined separately by Gross and Siebert and Abramovich and Chen. This paper provides a dictionary between log geometry and holomorphic exploded manifolds in order to compare Gromov-Witten invariants defined using exploded manifolds or log schemes. The gluing formula for Gromov-Witten invariants of exploded manifolds suggests an approach to proving analogous gluing formulas for log Gromov-Witten invariants. 相似文献
18.
19.
Using a generalized notion of symplectic Cayley transform in the symplectic group, we introduce a sequence of integer valued invariants (higher order signatures) associated with a degeneracy instant of a smooth path of symplectomorphisms. In the real analytic case, we give a formula for the Conley–Zehnder index in terms of the higher order signatures. 相似文献
20.
M.G. Sullivan 《Geometric And Functional Analysis》2002,12(4):810-872
This paper defines two K-theoretic invariants, Wh
1 and Wh
2, for individual and one-parameter families of Floer chain complexes. The chain complexes are generated by intersection points
of two Lagrangian submanifolds of a symplectic manifold, and the boundary maps are determined by holomorphic curves connecting
pairs of intersection points. The paper proves that Wh
1 and Wh
2 do not depend on the choice of almost complex structures and are invariant under Hamiltonian deformations. The proof of this
invariance uses properties of holomorphic curves, parametric gluing theorems, and a stabilization process.
Submitted: April 2001, Revised: December 2001, Final version: February 2002. 相似文献