首页/论文/2009.04897

arXiv 2009.048972020-09-09math.DGmath.DS

Analytic torsion, dynamical zeta function, and the Fried conjecture for admissible twists

Shu Shen

备注：: 40 pages. arXiv admin note: text overlap with arXiv:2009.03427
DOI：: 10.1007/s00220-021-04113-y

英文摘要：

We show an equality between the analytic torsion and the absolute value at the zero point of the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an acyclic flat vector bundle obtained by the restriction of a representation of the underlying Lie group. This generalises author's previous result for unitarily flat vector bundles, and the results of Bröcker, Müller, and Wotzke on closed hyperbolic manifolds.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Burghelea-Friedlander-Kappeler's gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion

The gluing formula of the zeta-determinant of a Laplacian given by Burghelea, Friedlander and Kappeler contains an unknown constant. In this paper we compute this constant to complete the formula under the assumption of the product structure near boundary. As applications of this result,we prove the adiabatic decomposition theorems of the zeta-determinant of a Laplacian with respect to the Dirichlet and Neumann boundary conditions and of the analytic torsion with respect to the absolute and relative boundary conditions.

2003-04-19math.DG

Analytic torsion, dynamical zeta functions, and the Fried conjecture

We prove the equality of the analytic torsion and the value at zero of a Ruelle dynamical zeta function associated with an acyclic unitarily flat vector bundle on a closed locally symmetric reductive manifold. This solves a conjecture of Fried. This article should be read in conjunction with an earlier paper by Moscovici and Stanton.

2017-09-25math.DG

扭曲的动力学ζ函数与弗里德猜想

Twisted dynamical zeta functions and the Fried's conjecture

本文综述了鲁埃勒和塞伯格的扭曲动力学ζ函数及弗里德猜想，基于作者在 Institut Henri Poincaré 的迷你课程内容。

2026-03-05math.DGmath.NT

On Fried's conjecture for compact hyperbolic manifolds

Fried's conjecture is concerned with the behavior of dynamical zeta functions at the origin. For compact hyperbolic manifolds, Fried proved that for an orthogonal acyclic representation of the fundamental group, the twisted Ruelle zeta function is holomorphic at $s=0$ and its value at $s=0$ equals the Ray-Singer analytic torsion. He also established a more general result for orthogonal representations, which are not acyclic. The purpose of the present paper is to extend Fried's result to arbitrary finite dimensional representations of the fundamental group. The Ray-Singer analytic torsion is r…

2020-05-04math.DGmath.SP

双曲度量附近普通Fried猜想的反例

Counterexamples to the ordinary Fried conjecture near hyperbolic metrics

Bocconi University（博科尼大学）；Paraform

该研究借助LLM辅助构造三维双曲度量附近普通Fried猜想的反例，证明非循环酉线特征趋于平凡特征时，Ruelle ζ函数零点模与解析挠率平方的比值趋于0。

2026-10-16math.DGmath.DSmath.SP

Spectral zeta function on discrete tori and Epstein-Riemann conjecture

We consider the combinatorial Laplacian on a sequence of discrete tori which approximate the m-dimensional torus. In the special case m=1, Friedli and Karlsson derived an asymptotic expansion of the corresponding spectral zeta function in the critical strip, as the approximation parameter goes to infinity. There, the authors have also formulated a conjecture on this asymptotics, that is equivalent to the Riemann conjecture. In this paper, inspired by the work of Friedli and Karlsson, we prove that a similar asymptotic expansion holds for m=2. Similar argument applies to higher dimensions as we…

2022-02-04math.DGmath.SP

更多相关论文收起更多论文

Fried conjecture in small dimensions

We study the twisted Ruelle zeta function $ζ_X(s)$ for smooth Anosov vector fields $X$ acting on flat vector bundles over smooth compact manifolds. In dimension $3$, we prove Fried conjecture, relating Reidemeister torsion and $ζ_X(0)$. In higher dimensions, we show more generally that $ζ_X(0)$ is locally constant with respect to the vector field $X$ under a spectral condition. As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic $3$-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flo…

2019-09-30math.DGmath.DSmath.SP

关于莫尔斯 - 斯梅尔流的弗里德猜想：射线 - 辛格与米尔诺度量综述

The Fried Conjecture for Morse-Smale Flows: A Survey on Ray-Singer and Milnor Metrics

University of Waterloo（滑铁卢大学）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）

综述紧致黎曼流形上莫尔斯 - 斯梅尔流的广义弗里德猜想，通过构建框架、刻画向量场等建立相关不变量，最终证明射线 - 辛格度量与米尔诺度量一致，解析挠率有特定等式关系。

2026-10-19math.DG

等变流的Ruelle动力zeta函数

A Ruelle dynamical zeta function for equivariant flows

针对光滑流形上具紧商的真群作用，基于Guillemin迹公式的等变推广定义了满足非退化条件的等变流的Ruelle动力ζ函数，探讨其与等变解析挠率的关联（等变Fried猜想）并给出多例计算，包括经典Ruelleζ函数未定义的情况。

2025-02-12math.DG

Weighted extremal Kähler metrics and the Einstein-Maxwell geometry of projective bundles

We study the existence of weighted extremal Kähler metrics in the sense of Apostolov-Calderbank-Gauduchon-Legendre and Lahdili on the total space of an admissible projective bundle over a Hodge Kähler manifold of constant scalar curvature. Admissible projective bundles have been defined by Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman, and they include the projective line bundles (Hwang-Singer) and their blow-downs (Koiso-Sakane), thus providing a most general setting for extending the existence theory for extremal Kähler metrics pioneered by a seminal construction of Calabi. We obtain a ge…

2018-08-08math.DG

紧致双曲轨道曲面上的扭曲 Ruelle zeta 函数与 Reidemeister-Turaev 挠率

The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister-Turaev torsion

该文研究紧致双曲轨道曲面上扭曲 Ruelle zeta 函数在零点取值与 Reidemeister-Turaev 挠率的关系，推广 Fried 猜想至非酉表示并解决相位符号问题，同时计算消失阶与首项系数。

2023-11-17math.DGmath.DSmath.GT

New proof of the Cheeger-Muller Theorem

We present a short analytic proof of the equality between the analytic and combinatorial torsion. We use the same approach as in the proof given by Burghelea, Friedlander and Kappeler, but avoid using the difficult Mayer-Vietoris type formula for the determinants of elliptic operators. Instead, we provide a direct way of analyzing the behaviour of the determinant of the Witten deformation of the Laplacian. In particular, we show that this determinant can be written as a sum of two terms, one of which has an asymptotic expansion with computable coefficients and the other is very simple (no zeta…

2002-04-18math-phmath.DGmath.MP

有限差分法在平坦表面上与平坦单位向量丛的结合

Finite difference method on flat surfaces with a flat unitary vector bundle

本文研究了半翻译表面与平坦单位向量丛的离散拉普拉斯算子谱与弗里德里克扩展谱的渐进行为，同时推导了近谐函数的哈纳克型估计，并探讨了其与生成树、生成森林及加权循环根生成森林数量的渐近展开关系。

2020-01-14math.DGmath.FAmath.SP

接触 Anosov 流的动力学挠率

Dynamical torsion for contact Anosov flows

本文引入“动力学挠率”这一新对象，证明其与 Turaev 挠率的比值在非循环表示空间上局部常数，并将 Fried 猜想从双曲 3-流形扩展至任意奇数维双曲流形。

2020-06-23math.DGmath.DSmath.GT