We discuss the adiabatic decomposition formula of the ζ-determinant of a Laplace type operator on... more We discuss the adiabatic decomposition formula of the ζ-determinant of a Laplace type operator on a closed manifold. We also analyze the adiabatic behavior of the ζ-determinant of a Dirichlet to Neumann operator. This analysis makes it possible to compare the adiabatic decomposition formula with the Meyer-Vietoris type formula for the ζ-determinant proved by Burghelea, Friedlander and Kappeler. As a byproduct of this comparison, we obtain the exact value of the local constant which appears in their formula for the case of Dirichlet boundary condition.
In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the d... more In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the decomposition problem for the (regularized) ζ-determinant of the Laplacian on a manifold with cylindrical end into the ζ-determinants of the Laplacians with Dirichlet conditions on the manifold with boundary and on the half infinite cylinder. We also compute all the contributions to this formula explicitly.
In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the d... more In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the decomposition problem for the (regularized) ζ-determinant of the Laplacian on a manifold with cylindrical end into the ζ-determinants of the Laplacians with Dirichlet conditions on the manifold with boundary and on the half infinite cylinder. We also compute all the contributions to this formula explicitly.
We use the branching rules on SU (3) to show that if two Aloff-Wallach spaces M k,l and M k ,l ar... more We use the branching rules on SU (3) to show that if two Aloff-Wallach spaces M k,l and M k ,l are isospectral for the Laplacian acting on smooth functions, they are isometric. We also show that 1 is the non-zero smallest eigenvalue among all Aloff-Wallach spaces and compute the multiplicities.
We define a Chern-Simons invariant for a certain class of infinite volume hyperbolic 3-manifolds.... more We define a Chern-Simons invariant for a certain class of infinite volume hyperbolic 3-manifolds. We then prove an expression relating the Bergman tau function on a cover of the Hurwitz space, to the lifting of the function F defined by Zograf on Teichmüller space, and another holomorphic function on the cover of the Hurwitz space which we introduce. If the point in cover of the Hurwitz space corresponds to a Riemann surface X, then this function is constructed from the renormalized volume and our Chern-Simons invariant for the bounding 3-manifold of X given by Schottky uniformization, together with a regularized Polyakov integral relating determinants of Laplacians on X in the hyperbolic and singular flat metrics. Combining this with a result of Kokotov and Korotkin, we obtain a similar expression for the isomonodromic tau function of Dubrovin. We also obtain a relation between the Chern-Simons invariant and the eta invariant of the bounding 3-manifold, with defect given by the phase of the Bergman tau function of X.
In this note we announce the adiabatic decomposition formula for the ζ-determinant of the Dirac L... more In this note we announce the adiabatic decomposition formula for the ζ-determinant of the Dirac Laplacian. Theorem 1.1 of this paper extends the result of our earlier work (see and ), which covered the case of the invertible tangential operator. The presence of the non-trivial kernel of the tangential operator requires careful analysis of the small eigenvalues of the Dirac Laplacian, which employs elements of scattering theory.
Electronic Research Announcements of the American Mathematical Society, 2005
In this note, we announce gluing and comparison formulas for the spectral invariants of Dirac typ... more In this note, we announce gluing and comparison formulas for the spectral invariants of Dirac type operators on compact manifolds and manifolds with cylindrical ends. We also explain the central ideas in their proofs.
In this paper we compute half density volumes of the irreducible SU (2)representation spaces for ... more In this paper we compute half density volumes of the irreducible SU (2)representation spaces for Seifert fibred manifolds and graph manifolds. These representation spaces are not discrete so that the half density derived from the Reidemeister torsion is used as a measure. As an application of our result we get the exact value of the Jeffrey-Weitsman-Witten invariant of a Seifert fibred manifold with non-discrete irreducible SU (2)-representation space. JINSUNG PARK asymptotic expansion of the Witten invariant of a 3-manifold. They define this invariant using the Reidemeister torsion as a half density measure of the irreducible SU (2)-representation space. Hence to compute this invariant we must compute the Reidemeister torsion completely including the determinant term of the homology which is the half density measure of the SU (2)-representation space. In this paper we call this invariant as the Jeffrey-Weitsman-Witten invariant. From the above motivations we could consider naturally that this invariant might be computed by the method of [W2]. The examples to which we apply the method of Witten are Seifert fibred manifolds and graph manifolds. This is because these manifolds are made from the trivial circle bundle over the Riemann surfaces by twisting finite fibers. So the method of Witten is applicable with some modification. To compute the half density derived from the Reidemeister torsion, we must compute both the scalar part and the determinant part of the Reidemeister torsion. The method to compute the scalar part comes from [F]. For Seifert fibred manifolds and graph manifolds this value gives the weight of the half density to each connected component of the irreducible SU (2)-representation space. So we combine two methods of [F] and [W2] to compute the half density volumes of the irreducible SU (2)-representation spaces for Seifert fibred manifolds and graph manifolds. Our computing method is 'to cut and to paste' with a topological view point. This can be comparable to the method of P.Kirk and E.Klassen to compute the Chern-Simons invariants for 3-manifolds [K,K]. We decompose the given manifolds into simple pieces which we can deal easily and then we glue the data of the decomposed pieces and investigate the gluing maps. The data of the pieces and the gluing maps gives the result that we want to get. Now we explain how this paper is organized. In the section 2 we study basic examples which are the building blocks of Seifert fibred manifolds and graph manifolds. We study the Reidemeister torsions and the SU (2)-representation spaces of these basic examples. In the section 3 we compute the scalar part of the Reidemeister torsions of Seifert fibred manifolds and graph manifolds for Ad-SU (2)representations. The computing method of the section 3 comes from [F]. This method is exactly 'to cut and to paste' so that we can apply the result of the section 2. In the section 4 we integrate the determinant term of the first homology part of the Reidemeister torsion over the irreducible SU (2)-representation space. The integration process of this section is also 'to cut and to paste'. We get the half density volume of the irreducible SU (2)-representation space by investigation of
The eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,ℝ) is defin... more The eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,ℝ) is defined through a regularized trace following Melrose. It reduces to the standard definition in terms of eigenvalues in the case of a totally non-trivial spin structure. When the S1-fibers are rescaled, the metric becomes of non-exact fibered-cusp type near the ends. We completely describe the continuous spectrum of the Dirac operator with respect to the rescaled metric and its dependence on the spin structure, and show that the adiabatic limit of the eta invariant is essentially the volume of the base hyperbolic Riemann surface with cusps, extending some of the results of Seade and Steer.
We discuss the adiabatic decomposition formula of the ζ-determinant of a Laplace type operator on... more We discuss the adiabatic decomposition formula of the ζ-determinant of a Laplace type operator on a closed manifold. We also analyze the adiabatic behavior of the ζ-determinant of a Dirichlet to Neumann operator. This analysis makes it possible to compare the adiabatic decomposition formula with the Meyer-Vietoris type formula for the ζ-determinant proved by Burghelea, Friedlander and Kappeler. As a byproduct of this comparison, we obtain the exact value of the local constant which appears in their formula for the case of Dirichlet boundary condition.
In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the d... more In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the decomposition problem for the (regularized) ζ-determinant of the Laplacian on a manifold with cylindrical end into the ζ-determinants of the Laplacians with Dirichlet conditions on the manifold with boundary and on the half infinite cylinder. We also compute all the contributions to this formula explicitly.
In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the d... more In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the decomposition problem for the (regularized) ζ-determinant of the Laplacian on a manifold with cylindrical end into the ζ-determinants of the Laplacians with Dirichlet conditions on the manifold with boundary and on the half infinite cylinder. We also compute all the contributions to this formula explicitly.
We use the branching rules on SU (3) to show that if two Aloff-Wallach spaces M k,l and M k ,l ar... more We use the branching rules on SU (3) to show that if two Aloff-Wallach spaces M k,l and M k ,l are isospectral for the Laplacian acting on smooth functions, they are isometric. We also show that 1 is the non-zero smallest eigenvalue among all Aloff-Wallach spaces and compute the multiplicities.
We define a Chern-Simons invariant for a certain class of infinite volume hyperbolic 3-manifolds.... more We define a Chern-Simons invariant for a certain class of infinite volume hyperbolic 3-manifolds. We then prove an expression relating the Bergman tau function on a cover of the Hurwitz space, to the lifting of the function F defined by Zograf on Teichmüller space, and another holomorphic function on the cover of the Hurwitz space which we introduce. If the point in cover of the Hurwitz space corresponds to a Riemann surface X, then this function is constructed from the renormalized volume and our Chern-Simons invariant for the bounding 3-manifold of X given by Schottky uniformization, together with a regularized Polyakov integral relating determinants of Laplacians on X in the hyperbolic and singular flat metrics. Combining this with a result of Kokotov and Korotkin, we obtain a similar expression for the isomonodromic tau function of Dubrovin. We also obtain a relation between the Chern-Simons invariant and the eta invariant of the bounding 3-manifold, with defect given by the phase of the Bergman tau function of X.
In this note we announce the adiabatic decomposition formula for the ζ-determinant of the Dirac L... more In this note we announce the adiabatic decomposition formula for the ζ-determinant of the Dirac Laplacian. Theorem 1.1 of this paper extends the result of our earlier work (see and ), which covered the case of the invertible tangential operator. The presence of the non-trivial kernel of the tangential operator requires careful analysis of the small eigenvalues of the Dirac Laplacian, which employs elements of scattering theory.
Electronic Research Announcements of the American Mathematical Society, 2005
In this note, we announce gluing and comparison formulas for the spectral invariants of Dirac typ... more In this note, we announce gluing and comparison formulas for the spectral invariants of Dirac type operators on compact manifolds and manifolds with cylindrical ends. We also explain the central ideas in their proofs.
In this paper we compute half density volumes of the irreducible SU (2)representation spaces for ... more In this paper we compute half density volumes of the irreducible SU (2)representation spaces for Seifert fibred manifolds and graph manifolds. These representation spaces are not discrete so that the half density derived from the Reidemeister torsion is used as a measure. As an application of our result we get the exact value of the Jeffrey-Weitsman-Witten invariant of a Seifert fibred manifold with non-discrete irreducible SU (2)-representation space. JINSUNG PARK asymptotic expansion of the Witten invariant of a 3-manifold. They define this invariant using the Reidemeister torsion as a half density measure of the irreducible SU (2)-representation space. Hence to compute this invariant we must compute the Reidemeister torsion completely including the determinant term of the homology which is the half density measure of the SU (2)-representation space. In this paper we call this invariant as the Jeffrey-Weitsman-Witten invariant. From the above motivations we could consider naturally that this invariant might be computed by the method of [W2]. The examples to which we apply the method of Witten are Seifert fibred manifolds and graph manifolds. This is because these manifolds are made from the trivial circle bundle over the Riemann surfaces by twisting finite fibers. So the method of Witten is applicable with some modification. To compute the half density derived from the Reidemeister torsion, we must compute both the scalar part and the determinant part of the Reidemeister torsion. The method to compute the scalar part comes from [F]. For Seifert fibred manifolds and graph manifolds this value gives the weight of the half density to each connected component of the irreducible SU (2)-representation space. So we combine two methods of [F] and [W2] to compute the half density volumes of the irreducible SU (2)-representation spaces for Seifert fibred manifolds and graph manifolds. Our computing method is 'to cut and to paste' with a topological view point. This can be comparable to the method of P.Kirk and E.Klassen to compute the Chern-Simons invariants for 3-manifolds [K,K]. We decompose the given manifolds into simple pieces which we can deal easily and then we glue the data of the decomposed pieces and investigate the gluing maps. The data of the pieces and the gluing maps gives the result that we want to get. Now we explain how this paper is organized. In the section 2 we study basic examples which are the building blocks of Seifert fibred manifolds and graph manifolds. We study the Reidemeister torsions and the SU (2)-representation spaces of these basic examples. In the section 3 we compute the scalar part of the Reidemeister torsions of Seifert fibred manifolds and graph manifolds for Ad-SU (2)representations. The computing method of the section 3 comes from [F]. This method is exactly 'to cut and to paste' so that we can apply the result of the section 2. In the section 4 we integrate the determinant term of the first homology part of the Reidemeister torsion over the irreducible SU (2)-representation space. The integration process of this section is also 'to cut and to paste'. We get the half density volume of the irreducible SU (2)-representation space by investigation of
The eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,ℝ) is defin... more The eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,ℝ) is defined through a regularized trace following Melrose. It reduces to the standard definition in terms of eigenvalues in the case of a totally non-trivial spin structure. When the S1-fibers are rescaled, the metric becomes of non-exact fibered-cusp type near the ends. We completely describe the continuous spectrum of the Dirac operator with respect to the rescaled metric and its dependence on the spin structure, and show that the adiabatic limit of the eta invariant is essentially the volume of the base hyperbolic Riemann surface with cusps, extending some of the results of Seade and Steer.
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