arXiv research

Study of torsion forms for positive line bundles.

Paper studies the full asymptotic torsion forms of flat bundles.

Analytic torsion expansions for symmetric and complex homogeneous spaces.

Formula derived for torsion of modified Dirac operator.

We calculate the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.

The purpose of this paper is first to give an asymptotic formula for the holomorphic analytic torsion forms of a fibration associated with increasing powers of a given line bundle. Secondly, we generalize this formula, thanks to the theory of Toeplitz operators, in the case where the powers of the line bundle is replac…

The paper solves heat kernel asymptotics on non-degenerate CR manifolds.

The purpose of this article is to study the asymptotic expansion of Ray-Singer analytic tosion associated with increasing powers p of a given positive line bundle. Here we prove that the asymptotic expansion associated to a manifold contains only the terms of the form $p^{n-i} \log p, p^{n-i}$ for $i$-natural. For the …

On an asymptotically conic manifold $(M,g)$, we analyze the asymptotics of the integral kernel of the resolvent $R_q(k):=(Δ_q+k^2)^{-1}$ of the Hodge Laplacian $Δ_q$ on $q$-forms as the spectral parameter $k$ approaches zero, assuming that 0 is not a resonance. The first application we give is an $L^p$ Sobolev estimate…

Explicit relation found between knot torsion and TQFT signatures.

New hyperbolic manifolds show exponential homology torsion growth.

We extend the complex-valued analytic torsion, introduced by Burghelea and Haller on closed manifolds, to compact Riemannian bordisms. We do so by considering a flat complex vector bundle over a compact Riemannian manifold, endowed with a fiberwise nondegenerate symmetric bilinear form. The Riemmanian metric and the bi…

Projective pre-compactness induces projective structure on boundary, relates to GR asymptotic forms.

We prove that for certain sequences of hyperbolic three--manifolds with cusps which converge to hyperbolic three--space in a weak ("Benjamini-Schramm") sense and certain coefficient systems the regularized analytic torsion approximates the $L^2$-torsion of the universal cover under an additional hypothesis. We also pro…

We study the analytic torsion of odd-dimensional hyperbolic orbifolds $Γ\backslash \mathbb{H}^{2n+1}$, depending on a representation of $Γ$. Our main goal is to understand the asymptotic behavior of the analytic torsion with respect to sequences of representations associated to rays of highest weights.

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n \ge 1$ with a transversal CR $S^1$-action on $X$. We introduce the Fourier components of the Ray-Singer analytic torsion on $X$ with respect to the $S^1$-action. We establish an asymptotic formula for the Fourier components of the an…

We show that a PSL(2;R)-representation of a Fuchsian group induces the asymptotics of the Reidemeister torsion for the Seifert manifold corresponding to the euler class of the PSL(2;R)-representation. We also show that the limit of leading coefficient of the Reidemeister torsion is determined by the euler class of a PS…

Paper compares absolute and relative real analytic torsion forms over fibrations.

New spectral torsion defined for rescaled Dirac operators.

Study on instantons in $G_2$ and $Spin(7)$ manifolds.

We review the Reidemeister torsion, Ray-Singer's analytic torsion and the Cheeger-M"uller theorem. We describe the analytic torsion of the de Rham complex twisted by a flux form introduced by the current authors and recall its properties. We define a new twisted analytic torsion for the complex of invariant differentia…

Analytic surgery and gluing formula for torsion forms in fiber bundles.

Simplified proof of gluing formula for analytic torsion forms.

Study curvature and torsion from cross-ratios in discrete curves.

We determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible SL(2;C)-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We al…

We introduce and study a canonical quadratic form, called the torsion quadratic form, of the determinant line of a flat vector bundle over a closed oriented odd-dimensional manifold. This quadratic form caries less information than the refined analytic torsion, introduced in our previous work, but is easier to construc…

We study the asymptotics of the higher dimensional Reidemeister torsion for torus knot exteriors, which is related to the results by W. Müller and P. Menal-Ferrer and J. Porti on the asymptotics of the Reidemeister torsion and the hyperbolic volumes for hyperbolic 3-manifolds. We show that the sequence of log |the high…

We introduce multi-torsion, a spectral invariant generalizing Ray-Singer analytic torsion. We define multi-torsion for compact manifolds with a certain local geometric product structure that gives a bigrading on differential forms. We prove that multi-torsion is metric-independent in a suitable sense. Our definition of…

The paper calculates the full asymptotics of analytic torsions for compact orbifolds.

We show that (under mild assumptions) the generating function of log homology torsion of a knot exterior has a meromorphic continuation to the entire complex plane. As corollaries, this gives new proofs of (a) the Silver-Williams asymptotic, (b) Fried's theorem on reconstructing the Alexander polynomial (c) Gordon's th…

Consider a flat vector bundle F over compact Riemannian manifold M and let f be a self-indexing Morse function on M. Let g be a smooth Euclidean metric on F. Set g_t=exp(-2tf)g and let ρ(t) be the Ray-Singer analytic torsion of F associated to the metric g_t. Assuming that the vector field $grad f$ satisfies the Morse-…

Paper introduces a new multilinear functional for spectral triples and computes its properties.

Formula for analytic torsion forms in fibrations by projective curves.

The paper analyzes the asymptotic behavior of a knot polynomial for a specific real number.

Study torsion and curvature in ACYT and AHKT 8-manifolds.

The paper studies curvature identities and solitons on Spin(7)-manifolds.

Manifolds with fibered cusps are a class of complete noncompact Riemannian manifolds including all locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asympt…

When does the amount of torsion in the homology of an arithmetic group grow exponentially with the covolume? We give many examples where this is so, and conjecture precise conditions.

New insights on ACYT 6-manifolds reveal instanton conditions for curvature.

This paper continues math.DG/9903140. Here we construct a linking form on the torsion part of middle dimensional extended L^2 homology and cohomology of odd-dimensional manifolds. We give a geometric necessary condition when this linking form is hyperbolic. We compute this linking form in case when the manifold bounds.…

For a compact manifold, which has a part isometric to a cylinder of finite length, we consider an adiabatic limit procedure, in which the length of the cylinder tends to infinity. We study the asymptotic of the spectrum of Hodge-Laplacian and the asymptotic of the $L^2$-metric on de Rham cohomology. As an application, …

We study the intrinsic torsion of almost quaternion-Hermitian manifolds via the exterior algebra. In particular, we show how it is determined by particular three-forms formed from simple combinations of the exterior derivatives of the local Kaehler forms. This gives a practical method to compute the intrinsic torsion a…

We calculate the twisted Reidemeister torsion of the complement of an iterated torus knot associated with a representation of its fundamental group to the complex special linear group of degree two. We also show that the twisted Reidemeister torsions associated with various representations appear in the asymptotic expa…

Bi-forms extend contrast functions to handle torsion in information geometry.

We compute the asymptotic dimension of the rationals given with an invariant proper metric. Also, we show that a countable torsion abelian group taken with an invariant proper metric has asymptotic dimension zero.

We define analytic torsion for the twisted de Rham complex, consisting of the spaces of differential forms on a compact oriented Riemannian manifold X valued in a flat vector bundle E, with a differential given by a flat connection on E plus an odd-degree closed differential form H on X. The difficulty lies in the fact…

We study how a gluing construction, which produces compact manifolds with holonomy G_2 from matching pairs of asymptotically cylindrical G_2-manifolds, behaves under deformations. We show that the gluing construction defines a smooth map from a moduli space of gluing data to the moduli space of torsion-free G_2-structu…

The study explores properties of metric connections with skew torsion and their curvature identities.