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Digitale Infotage für Schüler*innen vom 06.-09. Februar 2023

Photo: Universität Paderborn, Adelheid Rutenburges

Dr. Benjamin Delarue, (ehem. Küster)

Contact
Publications
Dr. Benjamin Delarue, (ehem. Küster)

Spectral Analysis

Research Associate - Principal Investigator, DFG project "Resonances for non-compact locally-symmetric spaces"

Office:
D2.207
Office hours:

By appointment

Web(external):
Visitor:
Warburger Str. 100
33098 Paderborn

Open list in Research Information System

2022

The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds

M. Cekić, B. Delarue, S. Dyatlov, G.P. Paternain, Inventiones mathematicae (2022), 229(1), pp. 303-394

<jats:title>Abstract</jats:title><jats:p>We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Sigma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> with Betti number <jats:inline-formula><jats:alternatives><jats:tex-math>$$b_1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula>, the order of vanishing of the Ruelle zeta function at zero equals <jats:inline-formula><jats:alternatives><jats:tex-math>$$4-b_1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, while in the hyperbolic case it is equal to <jats:inline-formula><jats:alternatives><jats:tex-math>$$4-2b_1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle <jats:inline-formula><jats:alternatives><jats:tex-math>$$S\Sigma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>Σ</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> with harmonic 1-forms on <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Sigma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>


2021

Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions

B. Delarue, P. Ramacher, Journal of Symplectic Geometry (2021), 19(6), pp. 1281 - 1337

DOI


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