Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps

For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of \(\mathrm{SL }_2(\mathbb {C})\) to the corresponding Reidemeister torsion. While the analytic torsion is a spectral invariant of the manifold, the Reidemeister torsion is of combinatorial nature. Our proof rests on an expression of the analytic torsion in terms of special values of Ruelle zeta functions as well as on recent work of Pere Menal-Ferrer and Joan Porti.     

Conformally Einstein products and nearly Kähler manifolds

In the first part of this note we study compact Riemannian manifolds (M, g) whose Riemannian product with is conformally Einstein. We then consider 6-dimensional almost Hermitian manifolds of type W ₁ + W ₄ in the Gray–Hervella classification admitting a parallel vector field and show that (under some mild assumption) they are obtained as Riemannian cylinders over compact Sasaki–Einstein 5-dimensional manifolds.      

Hankel operators on planar domains

Let Ω be a bounded domain in the plane whose boundary consists of a finite number of disjoint analytic simple closed curves LetA denote the space of analytic functions on Ω which are square integrable over Ω with respect to area measure and letP denote the orthogonal projection ofL ²(Ω,dA) ontoA. A functionb inA induces a Hankel operator (densely defined) onA by the ruleH _b (g)=(I?P)bg. This paper continues earlier investigations of the authors and others by determining conditions under whichH _b is bounded, compact, or lies in the Schatten-von Neumann idealS _p , 1<p<∞     

Reidemeister torsion and the K-theory of von Neumann algebras

We introduce a topological-type invariant for a cocompact properly discontinuous action of a discrete group on a Riemannian manifold generalizing classical notions of Reidemeister torsion. It takes values in the weak algebraic K-theory of the von Neumann algebra of . We give basic tools for its computation like sum and product formulas and calculate it in several cases. It encompasses, for instance, the Alexander polynomial and is related to analytic torsion.     

Magnetic Bi-harmonic differential operators on Riemannian manifolds and the separation problem

In this paper we obtain sufficient conditions for the bi-harmonic differential operator A = Δ_E² + q to be separated in the space L² (M) on a complete Riemannian manifold (M,g) with metric g, where Δ_E is the magnetic Laplacian onM and q ≥ 0 is a locally square integrable function on M. Recall that, in the terminology of Everitt and Giertz, the differential operator A is said to be separated in L² (M) if for all u ∈ L² (M) such that Au ∈ L² (M) we have Δ_E²u ∈ L² (M) and qu ∈ L² (M).     

$ L^2 $-torsion of Hyperbolic Manifolds of Finite Volume

Suppose is a compact connected odd-dimensional manifold with boundary, whose interior M comes with a complete hyperbolic metric of finite volume. We will show that the -topological torsion of and the -analytic torsion of the Riemannian manifold M are equal. In particular, the -topological torsion of is proportional to the hyperbolic volume of M, with a constant of proportionality which depends only on the dimension and which is known to be nonzero in odd dimensions [HS]. In dimension 3 this proves the conjecture [Lü2, Conjecture 2.3] or [LLü, Conjecture 7.7] which gives a complete calculation of the -topological torsion of compact -acyclic 3-manifolds which admit a geometric JSJT-decomposition.?In an appendix we give a counterexample to an extension of the Cheeger-Müller theorem to manifolds with boundary: if the metric is not a product near the boundary, in general analytic and topological torsion are not equal, even if the Euler characteristic of the boundary vanishes. Submitted: March 1998, revised: July 1998.     

Polynomials Annihilating the Witt Ring

Let F be a non-formally real field of characteristic not 2 and let W(F) be the Witt ring of F. In certain cases generators for the annihilator ideal are determined. Aim the primary decomposition of A(F) is given. For formally d fields F, as an analogue the primary decomposition of A_t(F) = {f(X) ∈ Z[X]| f(ω) = 0 for all ω ∈ W_t(F)}, where W_t(F) is the torsion part of the Witt group, is obtained.     

Connections on tangent bundles of higher order associated to regular Lagrangians

Given a regular Lagrangian L of order k on M we show that there exists a canonical connection _L on T ^2k–1 M whose paths are the solutions of the Euler-Lagrange equations for L. If L is the kinetic energy defined by a Riemannian metric then _L is the Riemannian connection.Partially supported by Subdirección General de Cooperación Internacional, Programa de Cooperación Científica con Iberoamérica, SPAIN and CNPq-BRAZIL, Proc. 30.1115/79.     

The center of the 3-dimensional and 4-dimensional Sklyanin algebras

LetA=A(E, ) denote either the 3-dimensional or 4-dimensional Sklyanin algebra associated to an elliptic curveE and a point E. Assume that the base field is algebraically closed, and that its characteristic does not divide the dimension ofA. It is known thatA is a finite module over its center if and only if is of finite order. Generators and defining relations for the centerZ(A) are given. IfS=Proj(Z(A)) andA is the sheaf ofO _S -algebras defined byA(S _(f))=A[f ^–1]₀ then the centerL ofA is described. For example, for the 3-dimensional Sklyanin algebra we obtain a new proof of M. Artin's result thatSpec L². However, for the 4-dimensional Sklyanin algebra there is not such a simple result: althoughSpec L is rational and normal, it is singular. We describe its singular locus, which is also the non-Azumaya locus ofA.     

Derived Picard Groups of Finite-Dimensional Hereditary Algebras

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPic _k (A) is the group of triangle auto-equivalences of D> ^b( mod A) induced by two-sided tilting complexes. We study the group DPic _k (A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic _k , as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic _k (A) on a certain infinite quiver ^irr. This representation is faithful when the quiver of A is a tree, and then DPic _k (A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic _k (A). When A is hereditary, DPic _k (A) coincides with the full group of k-linear triangle auto-equivalences of D^b( mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^b( mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.     

A class of Banach algebras whose duals have the Radon-Nikodym property

Let A be a complex, commutative Banach algebra and let M_A be the structure space of A. Assume that there exists a continuous homomorphism h : L¹(G) → A with dense range, where L¹(G) is the group algebra of a locally compact abelian group G. The main results of this paper can be summarized as follows: (a) If the dual space A* has the Radon-Nikodym property, then M_A is scattered (i.e., it has no nonempty perfect subset) and . (b) If the algebra A has an identity, then the space A* has the Radon-Nikodym property if and only if . Furthermore, any of these conditions implies that M_A is scattered. Several applications are given. Received: 29 September 2005     

The moduli space of complete embedded constant mean curvature surfaces

We examine the space of finite topology surfaces in ³ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the spaceM _k of all such surfaces withk ends (where surfaces are identified if they differ by an isometry of ³) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has noL ²-nullspace, we prove thatM _k is locally the quotient of a real analytic manifold of dimension 3k–6 by a finite group (i.e. a real analytic orbifold), fork 3. This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. It is of interest to note that the dimension ofM _k is independent of the genus of the underlying punctured Riemann surface to which is conformally equivalent. These results also apply to hypersurfaces of H ⁿ⁺¹ with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.Research of the first author supported in part by NSF grant # DMS9404278 and an NSF Postdoctoral Fellowship, of the second auther by NSF Young Investigator Award, a Sloan Foundation Postdoctoral Fellowship and NSF grant # DMS9303236, and of the third author by NSF grant # DMS9022140 and an NSF Postdoctoral Fellowship.     

Indecomposables as Elements in Affine Composition Algebras

LetAbe a path algebra of tame type over a finite field, letMbe an indecomposableA-module, and let (A) be the composition algebra ofA. The main result in this paper is that [M] ∈ (A) if and only ifMis a stone, i.e., Ext¹_A(M, M) = 0.     

Sum of weighted Lebesgue spaces and nonlinear elliptic equations

We study the sum of weighted Lebesgue spaces, by considering an abstract measure space (W,A,m){(\Omega ,\mathcal{A},\mu)} and investigating the main properties of both the Banach space

Hilbertian Versus Hilbert W*-Modules and Applications to L2- and other invariants

Hilbert(ian) A-modules over finite von Neumann algebras with a faithful normal trace state (from global analysis) and Hilbert W*-modules over A (from operator algebra theory) are compared and a categorical equivalence is established. The correspondence between these two structures sheds new light on basic results in L ²-invariant theory providing alternative proofs. We indicate new invariants for finitely generated projective B-modules, where B is a unital C*-algebra (usually the full group C*-algebra C*() of the fundamental group =₁(M) of a manifold M).     

Factorization properties of subdiagonal algebras

Let M be a von Neumann algebra equipped with a normal finite faithful normalized trace τ, and let A be a tracial subalgebra of M. Let E be a symmetric quasi-Banach space on [0, 1]. We show that A has an L_E(M)-factorization if and only if A is a subdiagonal algebra.     

Chem classes and Lie-Rinehart algebras

Let A be a F-algebra where F is a field, and let W be an A-module of finite presentation. We use the linear Lie-Rinehart algebra V_W of W to define the first Chern-class c₁(W) in , where U in Spec(A) is the open subset where W is locally free. We compute explicitly algebraic V_W-connections on maximal Cohen-Macaulay modules W on the hypersurface-singularities B_mn2 = x^m + yⁿ + z², and show that these connections are integrable, hence the first Chern-class c₁(W) vanishes. We also look at indecomposable maximal Cohen-Macaulay modules on quotient-singularities in dimension 2, and prove that their first Chern-class vanish.